Suppose f(x) is a function with continuous first and second derivatives on the closed interval [0,4] whose values for f and f' at x=0 and x=4 are given as
x=0; f(x)=3; f'(x)=3
x=4; f(x)=5; f'(x)=-1

Prove there exists a value of d, 0

Question

Suppose f(x) is a function with continuous first and second derivatives on the closed interval [0,4] whose values for f and f' at x=0 and x=4 are given as
x=0; f(x)=3; f'(x)=3
x=4; f(x)=5; f'(x)=-1

Prove there exists a value of d, 0<d<4, such that f''(d) = -1.

anonymous · Answer

isnt this suppose 2 b in da algebra section

anonymous · Answer

no. I believe it has to do with the Mean Value Theorem or Rolle's Theorem.

anonymous · Answer

it is mvt directly

anonymous · Answer

$$f'(0)=-1,f'(4)=3$$
$$\frac{f'(4)-f'(0)}{4-0}=\frac{-1-3}{4}=-1$$