Question

4. Consider the function g(x) = 2x^2 - 4x+3 on the interval [-1, 2]

Answer 1

Hi there!

A.) Begin by verifying that both endpoints have the same y-value:

g(-1) = 2(-1)² - 4(-1) + 3

Simplify:

g(-1) = 2 + 4 + 3 = 9

g(2) = 2(2)² - 4(2) + 3 = 8 - 8 + 3 = 3

Since the endpoints are not the same, Rolle's theorem does NOT apply.

B.)

Begin by ensuring that the function is continuous.

The function is a polynomial, so it satisfies the conditions of the function being BOTH continuous and differentiable on the given interval (All x-values do as well in this instance). We can proceed to find the values that satisfy the MVT:

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

Begin by finding the average rate of change over the interval:

$\frac{g(2) - g(-1)}{2-(-1)} = \frac{3 - 9 }{2-(-1)} = \frac{-6}{3} = -2$

Now, Find the derivative of the function:

g(x) = 2x² - 4x + 3

Apply power rule:

g'(x) = 4x - 4

Find the x value in which the derivative equals the AROC:

4x - 4 = -2

Add 4 to both sides:

4x = 2

Divide both sides by 4:

x = 1/2