Question

Determine whether the Mean Value Theorem for Derivatives applies for F(x)= x-cosx, on [x/2,-x/2] and find the coordinates of the points whose existence is guaranteed by the theorem

Answer 1

hmm what a weird looking interval you have

Answer 2

Scratch that make it [pi/2,-pi/2]

Answer 3

no such interval

Answer 4

[-pi/2,pi/2]

Answer 5

that makes sense
yes, the mvt applies because your function is continuous
how you are going to solve i have no idea, but
\[f(x)=x-\cos(x)\] so \(f(\frac{\pi}{2})=\frac{\pi}{2}-\cos(\frac{\pi}{2})=\frac{\pi}{2}-0=\frac{\pi}{2}\)

Answer 6

and \[f(-\frac{\pi}{2})=-\frac{\pi}{2}\] similar to above
therefore
\[\frac{f(\frac{\pi}{2})-f(-\frac{\pi}{2})}{\frac{\pi}{2}+\frac{\pi}{2}}\]
\[=\frac{(\frac{\pi}{2}+\frac{\pi}{2})}{\pi}=1\]

Answer 7

and \[f'(x)=1+\sin(x)\] so you need so solve
\[1+\sin(x)=1\] for \(x\) in the interval
should not be too hard actually

Answer 8

Right so I take the arcsin of 0 and get the answer?