f(x)=x2−2x−3x+2,[−1,3]f(x)=x2-2x-3x+2,[-1,3]

f(x)=x2−2x−3x+2,[−1,3]f(x)=x2-2x-3x+2,[-1,3]

  • MATH

g(x)=x2−2x−8g(x)=x2-2x-8 Identify the open intervals on which the function is increasing or…

Given: g(x)=x2−2x−8g(x)=x2-2x-8 Find the critical values by setting the derivative equal to zero and solving for the x value(s). g'(x)=2x−2=0g′(x)=2x-2=0 2x=22x=2 x=1x=1 The critical value is x=1. If g'(x)>0 the…

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  • MATH

f'(x)=6x−1,(2,7)f′(x)=6x-1,(2,7) Find a function ff that has the derivative f'(x)f′(x) and whose graph…

You need to notice that if the derivative is a linear function, then the primitive is a quadratic function, such that: f(x)=ax2+bx+cf(x)=ax2+bx+cDifferentiating f(x) yields: f'(x)=2ax+bf′(x)=2ax+b You need…

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  • MATH

f'(x)=2x,(1,0)f′(x)=2x,(1,0) Find a function ff that has the derivative f'(x)f′(x)and whose graph…

You need to notice that if the derivative is a linear function, then the primitive is a quadratic function, such that: f(x)=ax2+bx+cf(x)=ax2+bx+cDifferentiating f(x) yields: f'(x)=2ax+bf′(x)=2ax+b You need…

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  • MATH

f'(x)=4,(0,1)f′(x)=4,(0,1) Find a function ff that has the derivative f'(x)f′(x)and whose graph…

You need to notice that if the derivative is a constant function, then the primitive is a linear function, such that: f(x)=ax+bf(x)=ax+b Differentiating f(x) yields: f'(x)=af′(x)=a You need to set equal…

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  • MATH

f'(x)=0,(2,5)f′(x)=0,(2,5) Find a function ff that has the derivative f'(x)f′(x)and whose graph…

The function whose derivative is always 0 is a constant function. Given that at x=2 f(x)=5 we make a conclusion that this constant is 5. The answer: f(x)=5. (no, I cannot use 120 words here)

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  • MATH

2x−2−cos(x)=02x-2-cos(x)=0 Use the Intermediate Value Theorem and Rolle’s Theorem to prove that…

You need to evaluate the derivative of the function f(x)=2x−2−cosxf(x)=2x-2-cosx , such that: f'(x) = 2 + sin x You need to use Rolle’s theorem, so you need to find the roots of the equation 2 + sin x =…

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  • MATH

3x+1−sin(x)=03x+1-sin(x)=0 Use the Intermediate Value Theorem and Rolle’s Theorem to prove that…

Consider f(x)=3x+1-sinx. This function is continuous and infinitely differentiable on Rℝ. f(0)=1>0f(0)=1>0 and f(−π)=1−3π<0.f(-π)=1-3π<0. By the Intermediate Value Theorem there is at least one

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  • MATH

2×5+7x−1=02×5+7x-1=0 Use the Intermediate Value Theorem and Rolle’s Theorem to prove that…

f(x)=2×5+7x−1f(x)=2×5+7x-1 f(0)=2⋅05+7⋅0−1=−1f(0)=2⋅05+7⋅0-1=-1 f(1)=2⋅15+7⋅1−1=8f(1)=2⋅15+7⋅1-1=8 So,f(0) is negative and f(1) is positive. Since f(x) is continuous, by the Intermediate value theorem there is a number c between 0 and 1…

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  • MATH

x5+x3+x+1=0x5+x3+x+1=0 Use the Intermediate Value Theorem and Rolle’s Theorem to prove…

Consider x=-1 and x=0. For x=-1 f(x)<0 and for x=0 f(x)>0. By Intermediate Value Theorem there is at least one root of f at the interval (-1, 0). Let’s suppose that there are two roots. Then…

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  • MATH

f(x)=x4−2×3+x2,[0,6]f(x)=x4-2×3+x2,[0,6] Use a graphing utility to (a) graph the function ff on the…

Given f(x)=x4−2×3+x2f(x)=x4-2×3+x2 on the interval [0,6]: (1) This is a quartic polynomial with positive leading coefficient so its end behavior is the same as a parabola opening up. (2) f(0)=0 and…

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  • MATH

f(x)=x−−√,[1,9]f(x)=x,[1,9] Use a graphing utility to (a) graph the function ffon the given…

  1. a) the graph is here: https://www.desmos.com/calculator/kdzzqcpt3q b) the endpoints of the graph are (0, 0) and (9, 3). The slope of the secant line is 1/3, the equation is y=x/3. c)…

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  • MATH

f(x)=x−2sin(x),[−π,π]f(x)=x-2sin(x),[-π,π] Use a graphing utility to (a) graph the function ff on the…

f(-pi) = -pi, f(pi) = pi. The secant line goes through the points (−π,−π)(-π,-π) and (π,π)(π,π) . The equation is obviously y=x, its slope is 1. Let’s find points where f'(x)=1: f'(x) = 1 – 2cos(x)….

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  • MATH

f(x)=xx+1,[(−12),2]f(x)=xx+1,[(-12),2] Use a graphing utility to (a) graph the function ff on the…

(1) f(x)=xx+1f(x)=xx+1 is a rational function. It has a vertical asymptote at x=-1, and a horizontal asymptote of y=1. On the interval [-1/2,2] the graph is increasing towards the limiting value of…

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  • MATH

f(x)=cos(x)+tan(x),[0,π]f(x)=cos(x)+tan(x),[0,π] Determine whether the Mean Value Theorem can be applied to…

f(x)=cosx+tanxf(x)=cosx+tanx Mean value theorem can be applied, 1. if f is continuous on the closed interval [a,b][a,b] , 2. if f is differentiable on the open interval (a,b) 3. there is a number c in (a,b) such…

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  • MATH

f(x)=sin(x),[0,π]f(x)=sin(x),[0,π] Determine whether the Mean Value Theorem can be applied to ff on…

Yes, it can. The function f is contionuous on [0,π][0,π] and is differentiable on (0,π)(0,π), which are all the requirements for the Mean Value Theorem. Therefore there is at least one point c on…

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  • MATH

f(x)=2−x−−−−−√,[−7,2]f(x)=2-x,[-7,2] Determine whether the Mean Value Theorem can be applied to ff…

The mean value theorem is applicable to the given function, since it is a polynomial function. All polynomial functions are continuous and differentiable on R, hence, the given function is…

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  • MATH

f(x)=|2x+1|,[−1,3]f(x)=|2x+1|,[-1,3] Determine whether the Mean Value Theorem can be applied to ff on…

NO, it isn’t. The function is continuous on [-1, 3] and is differentiable on (-1, 3) but the point where 2x+1=0, or x=-1/2 in [-1, 3]. To the left of this point f(x)=-2x-1, to the right f(x)=2x+1,…

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  • MATH

f(x)=x+1x,[−1,2]f(x)=x+1x,[-1,2] Determine whether the Mean Value Theorem can be applied to ff on…

The mean value theorem is applicable to the given function, since it is a polynomial function. All polynomial functions are continuous and differentiable on R, hence, the given function is…

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  • MATH

f(x)=x23,[0,1]f(x)=x23,[0,1] Determine whether the Mean Value Theorem can be applied to ff on…

The mean value theorem is applicable to the given function, since it is a polynomial function. All polynomial functions are continuous and differentiable on R, hence, the given function is…

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  • MATH

f(x)=x4−8x,[0,2]f(x)=x4-8x,[0,2] Determine whether the Mean Value Theorem can be applied to ff on…

The mean value theorem may be applied to the given function since all polynomial functions are continuous and differentiable on R, hence, the given function is continuous on [0,2] and…

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  • MATH

f(x)=x3+2x,[−1,1]f(x)=x3+2x,[-1,1] Determine whether the Mean Value Theorem can be applied to ff on…

Yes, it can. The function is continuous on [-1, 1] and is differentiable on (-1, 1). Here a=-1 and b=1. f(a) = f(-1) = -3 and f(b) = f(1) = 3. So f(b)−f(a)b−a=62=3.f(b)-f(a)b-a=62=3. f'(x) = 3x^2 +…

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  • MATH

f(x)=2×3,[0,6]f(x)=2×3,[0,6] Determine whether the Mean Value Theorem can be applied to ff on the…

The mean value theorem is applicable to the given function, since it is a polynomial function. All polynomial functions are continuous and differentiable on R, hence, the given function is…

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  • MATH

f(x)=x2,[−2,1]f(x)=x2,[-2,1] Determine whether the Mean Value Theorem can be applied to ff on the…

The mean value theorem is applicable to the given function, since it is a polynomial function. All polynomial functions are continuous and differentiable on R, hence, the given function is…

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  • MATH

f(x)=(x2)−sin(πx6),[−1,0]f(x)=(x2)-sin(πx6),[-1,0] Determine whether Rolle’s Theorem can be applied to ff on…

The given function is continuous and differentiable over the given interval, as all trigonometric functions are. For Rolle’s Theorem to be applied, you also need to test if f(−1)=f(0).f(-1)=f(0).

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  • MATH

f(x)=x−tan(πx),[−1414]f(x)=x-tan(πx),[-1414] Use a graphing utility to graph the function on the…

NO, Rolle’s Theorem isn’t applicable because the function doesn’t has equal values at the endpoints: f(1/4) = 1/4 – tan(pi/4) = 1/4 – 1 = -3/4 while f(-1/4) = -1/4 – tan(-pi/4) = -1/4 + 1 = 3/4.

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  • MATH

f(x)=x−x13,[0,1]f(x)=x-x13,[0,1] Determine whether Rolle’s Theorem can be applied to ff on the…

You need to notice that the given function is continuous on [0,1] and differentiable on (0,1), since it is a polynomial function. You need to verify if f(0)=f(1), hence, you need to evaluate the…

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  • MATH

f(x)=|x|−1,[−1,1]f(x)=|x|-1,[-1,1] Use a graphing utility to graph the function on the closed interval…

Rolle’s theorem cannot be applied because the function is not differentiable over the whole interval (−1,1).(-1,1). More specifically the function is not differentiable at zero. Graph of the function…

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  • MATH

f(x)=sec(x),[π,2π]f(x)=sec(x),[π,2π] Determine whether Rolle’s Theorem can be applied to ff on the…

The Rolle’s theorem is applicable to the given function, only if the function is continuous and differentiable over the interval, and f(a) = f(b). Since all trigonometric functions are continuous…

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  • MATH

f(x)=tan(x),[0,π]f(x)=tan(x),[0,π] Determine whether Rolle’s Theorem can be applied to ff on the…

  1. This function isn’t defined on entire interval (the point where it isn’t defined is pi/2). Actually, not only Rolle’s Theorem isn’t applicable but also its conclusion doesn’t hold: there is no…

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  • MATH

f(x)=cos(2x),[−π,π]f(x)=cos(2x),[-π,π] Determine whether Rolle’s Theorem can be applied to ff on the…

Rolle’s Theorem requires f to be defined and continuous on the given closed interval, differentiable on the open interval and values of f on ends to be equal. Here all conditions are met…

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  • MATH

f(x)=sin(3x),[0,(π3)]f(x)=sin(3x),[0,(π3)] Determine whether Rolle’s Theorem can be applied to ff on…

The Rolle’s theorem is applicable to the given function, only if the function is continuous and differentiable over the interval, and f(a) = f(b). Since all polynomial functions are continuous and…

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  • MATH

f(x)=cos(x),[0,2π]f(x)=cos(x),[0,2π] Determine whether Rolle’s Theorem can be applied to ff on the…

The Rolle’s theorem is applicable to the given function, only if the function is continuous and differentiable over the interval, and f(a) = f(b). Since all trigonometric functions are continuous…

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  • MATH

f(x)=sin(x),[0,2π]f(x)=sin(x),[0,2π] Determine whether Rolle’s Theorem can be applied to ff on the…

Yes, it can. Function f is continuous on [0,2π][0,2π] and is differentiable on(0,2π)(0,2π) . Also, f(0)=f(2π)f(0)=f(2π) (both =0). There are all conditions of Rolle’s Theorem. Because of this there is at…

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  • MATH

f(x)=x2−1x,[−1,1]f(x)=x2-1x,[-1,1] Determine whether Rolle’s Theorem can be applied to ff on…

The Rolle’s theorem is applicable to the given function, only if the function is continuous and differentiable over the interval, and f(a) = f(b). Since all polynomial functions are continuous and…

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  • MATH

f(x)=x2−2x−3x+2,[−1,3]f(x)=x2-2x-3x+2,[-1,3] Determine whether Rolle’s Theorem can be applied…

The Rolle’s theorem is applicable to the given function, only if the function is continuous and differentiable over the interval, and f(a) = f(b). Since all polynomial functions are continuous and…

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  • MATH

f(x)=3−|x−3|,[0,6]f(x)=3-|x-3|,[0,6] Determine whether Rolle’s Theorem can be applied to ff on the…

Given f(x)=3-|x-3| on the interval [0,6]: To apply Rolle’s theorem the function must be continuous on the closed interval [a,b], differentiable on the open interval (a,b), and f(a)=f(b). f is…

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  • MATH

f(x)=x23−1,[−8,8]f(x)=x23-1,[-8,8] Determine whether Rolle’s Theorem can be applied to ff on…

Given: f(x)=x^(2/3)-1,[-8,8]. Rolle’s Theorem does not apply to the function f(x) on the given interval because all 3 conditions of Rolle’s Theorem will not be met. The function f(x) is continuous…

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  • MATH

f(x)=(x−4)(x+2)2,[−2,4]f(x)=(x-4)(x+2)2,[-2,4] Determine whether Rolle’s Theorem can be applied to ff…

The Rolle’s theorem is applicable to the given function, only if the function is continuous and differentiable over the interval, and f(a) = f(b). Since all polynomial functions are continuous and…

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  • MATH

f(x)=(x−1)(x−2)(x−3),[1,3]f(x)=(x-1)(x-2)(x-3),[1,3] Determine whether Rolle’s Theorem can be applied to…

Yes it can. The function f is continuous on [1, 3] and differentiable on (1, 3) (as an elementary function, or even as polynomial) and f(1) = f(3) (=0). All conditions are met. The value of c is…

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  • MATH

f(x)=x2−8x+5,[2,6]f(x)=x2-8x+5,[2,6] Determine whether Rolle’s Theorem can be applied to ff on the…

Given: f(x)=x2−8x+5,[2,6].f(x)=x2-8x+5,[2,6]. Rolle’s Theorem can be applied because the function f(x) is a continuous polynomial on the closed interval [2,6] and differentiable on the open interval (2,6)., and…

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  • MATH

f(x)=−x2+3x,[0,3]f(x)=-x2+3x,[0,3] Determine whether Rolle’s Theorem can be applied to ff on the…

The Rolle’s theorem is applicable to the given function, only if the function is continuous and differentiable over the interval, and f(a) = f(b). Since all polynomial functions are continuous and…

1 educator answer

  • MATH

f(x)=−3xx+1−−−−−√f(x)=-3xx+1 Find the two x-intercepts of the function ffand show that 

Hello! f(x) = 0 for x=0 and x=-1 (obvious). f is continuous on [-1, 0] and is differentiable on (-1, 0) (as an elementary function). So we can aplly Rolle’s Theorem and conclude that there is at…

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  • MATH

f(x)=xx+4−−−−−√f(x)=xx+4 Find the two x-intercepts of the function ff and show that 

You need to find the two x intercepts of the function, hence, you need to solve for x the equation f(x) = 0, such that: f(x)=x⋅x+4−−−−−√=0f(x)=x⋅x+4=0 x⋅x+4−−−−−√=0x=0x⋅x+4=0x=0

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  • MATH

f(x)=x2+6xf(x)=x2+6x Find the two x-intercepts of the function ff and show that f'(x)=0f′(x)=0 at…

You need to find the two x intercepts of the function, hence, you need to solve for x the equation f(x) = 0, such that: f(x)=x2+6x=0f(x)=x2+6x=0You need to factor out x, such that:

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  • MATH

f(x)=x2−x−2f(x)=x2-x-2 Find the two x-intercepts of the function ff and show that f'(x)=0f′(x)=0…

The x-intercepts are roots of f(x)=0. There are only two of those: -1 and 2. The point between them where f'(x)=0 exists by Rolle’s Theorem and we can find it: f'(x)=2x-1, it is zero only at x=1/2.

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  • MATH

f(x)=((2−x23)3),[−1,1]−−−−−−−−−−−−−−−−−−√f(x)=((2-x23)3),[-1,1] Explain why Rolle’s Theorem does not apply to the…

You need to notice that the given function is continuous on [-1,1] and differentiable on (-1,1), since it is a polynomial function. You need to verify if f(−1)=f(1),f(-1)=f(1), hence, you need to evaluate…

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  • MATH

f(x)=1−|x−1|,[0,2]f(x)=1-|x-1|,[0,2] Explain why Rolle’s Theorem does not apply to the function…

This function satisfies some conditions of Rolle’s Theorem: it is continuous on [0, 2], differentiable almost everywhere on (0, 2) and f(0) = f(2) (=0). But there is one point where f isn’t…

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  • MATH

f(x)=cot(x2),[π,3π]f(x)=cot(x2),[π,3π] Explain why Rolle’s Theorem does not apply to the function…

The Rolle’s theorem is applicable to the given function, only if the function is continuous and differentiable over the interval, and f(a) = f(b). Since all trigonometric functions are continuous…

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  • MATH

f(x)=∣∣∣1x∣∣∣,[−1,1]f(x)=|1x|,[-1,1] Explain why Rolle’s Theorem does not apply to the function even…

Rolle’s Theorem requires the function to be continuous on the closed interval [a, b]. But this function isn’t. The problem point is x=0. Aclually, f(x) isn’t even defined at x=0, and because…

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  • MATH

f(x)=(43)x3−x−−−−−√,[0,3]f(x)=(43)x3-x,[0,3] (a) Use a computer algebra system to graph the function…

See the attached graph plotted in Matlab. a) approximate exterma There is a maximum close to 2. b) >> f = (4/3)*x*sqrt(3-x) f = (4*x*(3 – x)^(1/2))/3 1. Asymptotes – >> limit (f, inf)…

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