Locate the absolute value extrema of the function f(x)=cos(pi x) on the closed interval [0,3/4]

Determine whether Rolle's Theorem can be applied to the function f(x)=(x+3)(x-2)^2 on the closed interval [-3,2] if rolle's Theorem can be applied find all numbers c in the open interval (-3.2) such that f' (c)=0

Determine whether the Mean Value Theorem can be applied to the function f(x)=2 sin x + sin 2x on the closed interval [7 pi, 8 pi] If the mean value theorem can be applied find all numbers c in the open interval (7 pi, 8 pi) such that f'(c)= f(8 pi)-f(7 pi)/8 pi-7 pi

Question

Locate the absolute value extrema of the function f(x)=cos(pi x) on the closed interval [0,3/4]

Determine whether Rolle's Theorem can be applied to the function f(x)=(x+3)(x-2)^2 on the closed interval [-3,2] if rolle's Theorem can be applied find all numbers c in the open interval (-3.2) such that f' (c)=0

Determine whether the Mean Value Theorem can be applied to the function f(x)=2 sin x + sin 2x on the closed interval [7 pi, 8 pi] If the mean value theorem can be applied find all numbers c in the open interval (7 pi, 8 pi) such that f'(c)= f(8 pi)-f(7 pi)/8 pi-7 pi

thecalchater · Answer

@satellite73

thecalchater · Answer

@ganeshie8

thecalchater · Answer

I figured our rolles theorem but not the MVT i got the derivative and then plugged in 7pi and 8pi and got 0 where do i go from there?

thecalchater · Answer

@ganeshie8 @TheSmartOne

anonymous · Answer

For the first, you can compute the derivative, find the critical points, and apply the derivative test:
$$f(x)=\cos\pi x\implies f'(x)=-\pi\sin\pi x=0~~\implies \pi x=n\pi\implies x=n$$where $n$ is any integer. How many integers exist in the interval $\left[0,\dfrac{3}{4}\right]$?

For the second, $f(x)$ must be continuous and differentiable over the given interval $(a,b)$, with $f(a)=f(b)$. Since this is true, this guarantees the existence of $c$ such that
$$f'(c)=(c-2)^2+2(c+3)(c-2)=0\implies c=\cdots$$

For the third, the same conditions must hold for $f(x)$, except that $f(a)=f(b)$. Since $\sin x$ is a continuous function and any linear combination of continuous functions must also be continuous, you know that $f(x)$ satisfies the MVT. Now you want to find $c$ such that
$$f'(c)=\frac{f(8\pi)-f(7\pi)}{8\pi-7\pi}\implies 2\cos c+2\cos2c=0$$To solve for $c$, use an identity:
$$\cos2x=\cos^2x-\sin^2x=2\cos^2x-1$$So you have this equation with the LHS quadratic in $\cos c$:
$$2\cos c+2\cos2c=4\cos^2c+2\cos c-2=0$$

thomas5267 · Answer

Why does the second question requires the Rolle's theorem.  Since [-3,2] is closed and bounded and the f(x) in second question is continuous, just use the extreme value theorem.