Question

Consider the given function and the given interval. f(x) = (x − 4)2, [3, 6]
(a) Find the average value fave of f on the given interval. fave =
(b) Find c such that fave = f(c). c = (smaller value) c = (larger value)

Answer 1

Answer:

f average = 1

smaller value c = 3

larger value c = 5

Step-by-step explanation:

Answer 2

Functions can be represented using equations.

• The average value of the function is 1
• The smaller and larger values of c are: 3 and 5

The function is given as:

$\mathbf{f(x) = (x - 4)^2\ [3,6]}$

(a) The average value

This is calculated as:

$\mathbf{f_{ave} = \frac{f(b) - f(a)}{b - a}}$

So, we have:

$\mathbf{f_{ave} = \frac{f(6) - f(3)}{6 - 3}}$

$\mathbf{f_{ave} = \frac{f(6) - f(3)}{3}}$

Calculate f(6) and f(3)

$\mathbf{f(6) = (6 - 4)^2 = 4}$

$\mathbf{f(3) = (3 - 4)^2 = 1}$

So, we have:

$\mathbf{f_{ave} = \frac{4 - 1}{3}}$

$\mathbf{f_{ave} = \frac{3}{3}}$

$\mathbf{f_{ave} = 1}$

Hence, the average value of the function is 1

$\mathbf{(b)\ f_{ave} = f(c)}$

Substitute c for x in:

$\mathbf{f(x) = (x - 4)^2}$

$\mathbf{f(c) = (c - 4)^2}$

$\mathbf{f_{ave} = f(c) = 1}$

So, we have:

$\mathbf{ (c - 4)^2 = 1}$

Take square roots of both sides

$\mathbf{c - 4 = \pm 1}$

Add 4 to both sides

$\mathbf{c =4 \pm 1}$

Split

$\mathbf{c =(4 - 1,4+1)}$

$\mathbf{c =(3,5)}$

Hence, the smaller and larger values of c are: 3 and 5

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