Question

on which interval does the function ( g(x) = 4x - 3x^2 ) have an average rate of change equal to 16?

( x = -4 ) to ( x = 0 ) ( x = -4 ) to ( x = 1 )

( x = -4 ) to ( x = 2 ) ( x = -4 ) to ( x = 3 )

Explanation:

Step1: Recall the average rate of change formula

The average rate of change of a function \( g(x) \) over the interval \([a, b]\) is given by \(\frac{g(b) - g(a)}{b - a}\). We need to find the interval \([-4, b]\) (since all intervals start at \( x = -4 \)) such that \(\frac{g(b) - g(-4)}{b - (-4)} = 16\), where \( g(x)=4x - 3x^{2}\).

First, calculate \( g(-4) \):

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Step2: Substitute into the average rate of change formula

For a general \( b \), the average rate of change is \(\frac{g(b)-g(-4)}{b + 4}=\frac{(4b - 3b^{2})-(-64)}{b + 4}=\frac{4b - 3b^{2}+ 64}{b + 4}\). We set this equal to 16:

$$ \frac{4b - 3b^{2}+ 64}{b + 4}=16 $$

Multiply both sides by \( b + 4 \) (assuming \( b
eq - 4 \)):

$$ 4b - 3b^{2}+ 64 = 16(b + 4) $$

Expand the right - hand side:

$$ 4b - 3b^{2}+ 64 = 16b+64 $$

Subtract \( 16b + 64 \) from both sides:

$$ 4b - 3b^{2}+ 64-(16b + 64)=0 $$

$$ 4b - 3b^{2}+ 64 - 16b - 64 = 0 $$

Simplify the left - hand side:

$$ -3b^{2}-12b = 0 $$

Factor out \(-3b\):

$$ -3b(b + 4)=0 $$

The solutions are \( b = 0 \) or \( b=-4 \). But \( b=-4 \) makes the denominator of the average rate of change formula zero, so we discard it. So we check the interval \( x=-4 \) to \( x = 0 \).

Let's verify:
\( g(0)=4(0)-3(0)^{2}=0 \)
The average rate of change from \( x=-4 \) to \( x = 0 \) is \(\frac{g(0)-g(-4)}{0-(-4)}=\frac{0 - (-64)}{4}=\frac{64}{4}=16\)

Now let's check the other intervals for completeness:

• For \( b = 1 \):

\( g(1)=4(1)-3(1)^{2}=4 - 3 = 1 \)
Average rate of change: \(\frac{1-(-64)}{1 + 4}=\frac{65}{5}=13
eq16\)

• For \( b = 2 \):

\( g(2)=4(2)-3(2)^{2}=8 - 12=-4 \)
Average rate of change: \(\frac{-4-(-64)}{2 + 4}=\frac{60}{6}=10
eq16\)

• For \( b = 3 \):

\( g(3)=4(3)-3(3)^{2}=12 - 27=-15 \)
Average rate of change: \(\frac{-15-(-64)}{3 + 4}=\frac{49}{7}=7
eq16\)

Answer:

\( x=-4 \) to \( x = 0 \)