Question

Given the function `f(x) = 4 x`, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [4,9] and find the c?

Answer 1

General Solution Method

The Mean Value Theorem has two hypotheses:

H1 : `f` is continuous on the closed interval `[a,b]`

H2 : `f` is differentiable on the open interval `(a,b)`.

In this question, `f(x) = 4x` , `a=4` and `b=9`.

This function is a polynomial, so it is continuous on its domain, `(-oo, oo)`, so it is continuous on `[11, 23]`

`f'(x)= 4` which exists for all `x` (for all `x in (4,9)`,

So `f` is differentiable on `(4, 9)`

Therefore the hypotheses of The Mean Value Theorem are true for this function on this interval.

In general, to find `c`, solve `f'(x) = (f(b)-f(a))/(b-a)`, keeping only the solutions in `(a,b)`.

In this question we need to solve `4 = (4(9)-4(4))/(9-4) = 20/5 = 4`.

Every real number is a solution, so for `c` we may take any number in `(4,9)`.

Alternative solution for a linear function (like this one).

`f(x) = 4x` is a linear function, so the secant line joining the endpoints on the interval is exactly the graph of the function. It has slope `4`.

The `c` is the value at which the tangent line has the same slope as the secant line joining the endpoints. But at every number, the tangent line has slope `4`, so any number in the open interval will work for `c`.