Question

Suppose that 4 ≤ f '(x) ≤ 5 for all values of x. What are the minimum and maximum possible values of f(4) − f(2)? ______≤ f(4) − f(2) ≤ ______

Answer 1

Answer:

$8 \le f(4) - f(2) \le 10$

Step-by-step explanation:

Given

$4 \le f'(x) \le 5$

Required

Determine the minimum and maximum value of f(4) - f(2)

This question will be answered using the following mean value theorem.

$f'(x) = \frac{f(b) - f(a)}{b - a}$

In this case: $b = 4$ and $a = 2$

So, we have:

$f'(x) = \frac{f(4) - f(2)}{4 - 2}$

$f'(x) = \frac{f(4) - f(2)}{2}$

Make f(4) - f(2), the subject:

$f(4) -f(2) = 2 * f'(x)$

From the given range: $4 \le f'(x) \le 5$

We have that; f'(x) is at minimum at 4

So, the minimum of f(4) - f(2) is:

$f(4) -f(2) = 2 * 4$

$f(4) -f(2) = 8$

We have that; f'(x) is at maximum at 5

So, the maximum of f(4) - f(2) is:

$f(4) - f(2) = 2 * 5$

$f(4) - f(2) = 10$

Hence, the minimum and maximum values can be expressed as:

$8 \le f(4) - f(2) \le 10$