# 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…

1 educator answer

**MATH**

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…

1 educator answer

**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…

1 educator answer

**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…

1 educator answer

**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)

1 educator answer

**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 =…

1 educator answer

**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

1 educator answer

**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…

1 educator answer

**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…

1 educator answer

**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…

1 educator answer

**MATH**

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

- 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)…

1 educator answer

**MATH**

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)….

1 educator answer

**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…

1 educator answer

**MATH**

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…

1 educator answer

**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…

2 educator answers

**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…

1 educator answer

**MATH**

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,…

1 educator answer

**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…

1 educator answer

**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…

1 educator answer

**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…

1 educator answer

**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 +…

1 educator answer

**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…**

1 educator answer

**MATH**

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

1 educator answer

**MATH**

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).

1 educator answer

**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.

2 educator answers

**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…

1 educator answer

**MATH**

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…

1 educator answer

**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…

1 educator answer

**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…**

- 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…

1 educator answer

**MATH**

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…

1 educator answer

**MATH**

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)=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…

1 educator answer

**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…

1 educator answer

**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…

1 educator answer

**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…

1 educator answer

**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…

1 educator answer

**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…

1 educator answer

**MATH**

1 educator answer

**MATH**

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…

1 educator answer

**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…

1 educator answer

**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…**

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…

1 educator answer

**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

1 educator answer

**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:

1 educator answer

**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.

1 educator answer

**MATH**

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…

1 educator answer

**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…

1 educator answer

**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…

1 educator answer

**MATH**

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…

1 educator answer

**MATH**

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)…

1 educator answer