Question

Determine if the function f(x)=4x√−3x satisfies the Mean Value Theorem on [1, 25]. If so, find all numbers c on the interval that satisfy the theorem.

Answer 1

Find f'(x)
Find the slope of the secant
Set, f'(x)=slope of secant

Answer 2

Find:
f(1) and f(5),
and then the slope between x=1 and x=5.

Answer 3

Differentiate the f'(x).

Answer 4

Set f'(x)=slope between x=1 and x=5.

Answer 5

want example?

Answer 6

\(\color{#000000 }{ \displaystyle f(x)=4x^2-6x }\), on \([2,10]\).

\(\color{#000000 }{ \displaystyle f(2)=4 }\)
\(\color{#000000 }{ \displaystyle f(10)=340 }\)

\(\color{#000000 }{ \displaystyle {\rm
Slope }=\frac{\Delta y}{\Delta x} =\frac{f(10)-f(2)}{10-2} }\)

\(\color{#000000 }{ \displaystyle {\rm
Slope }=\frac{340-4}{8} =\frac{336}{8} =42}\)

That was the "slope of the secant".

Now the derivative-slope;
\(\color{#000000 }{ \displaystyle f'(x)=(2)4x^{2-1}-(1)6x^{1-1} }\)
\(\color{#000000 }{ \displaystyle f'(x)=8x-6}\) this is your slope-generator. (right?)

So, you want to find the points on the function that will have the same slope as the slope of the secant.

\(\color{#000000 }{ \displaystyle 42=8x-6}\)
\(\color{#000000 }{ \displaystyle x=6}\) (this is the conclusion of the MVT)