Question

which of the following functions have an average rate of change that is negative on the interval from x = -1 to x = 2? select all that apply.
$f(x) = x^2 + 3x + 5$
$f(x) = x^2 - 3x - 5$
$f(x) = 3x^2 - 5x$
$f(x) = 3x + 5x^2$

Explanation:

Step1: Recall Average Rate of Change Formula

The average rate of change of a function \( f(x) \) on the interval \([a, b]\) is given by \( \frac{f(b) - f(a)}{b - a} \). Here, \( a=-1 \) and \( b = 2 \), so the average rate of change is \( \frac{f(2)-f(-1)}{2-(-1)}=\frac{f(2)-f(-1)}{3} \). We need to check for each function if \( f(2)-f(-1)<0 \) (since the denominator 3 is positive, the sign of the average rate of change depends on the numerator).

Step2: Analyze \( f(x)=x^{2}+3x + 5 \)

• Calculate \( f(2) \): \( f(2)=2^{2}+3(2)+5=4 + 6+5 = 15 \)
• Calculate \( f(-1) \): \( f(-1)=(-1)^{2}+3(-1)+5=1-3 + 5=3 \)
• \( f(2)-f(-1)=15 - 3=12>0 \). So average rate of change is positive.

Step3: Analyze \( f(x)=x^{2}-3x - 5 \)

• Calculate \( f(2) \): \( f(2)=2^{2}-3(2)-5=4-6 - 5=-7 \)
• Calculate \( f(-1) \): \( f(-1)=(-1)^{2}-3(-1)-5=1 + 3-5=-1 \)
• \( f(2)-f(-1)=-7-(-1)=-6<0 \). So average rate of change is negative.

Step4: Analyze \( f(x)=3x^{2}-5x \)

• Calculate \( f(2) \): \( f(2)=3(2)^{2}-5(2)=12 - 10 = 2 \)
• Calculate \( f(-1) \): \( f(-1)=3(-1)^{2}-5(-1)=3 + 5 = 8 \)
• \( f(2)-f(-1)=2 - 8=-6<0 \). So average rate of change is negative.

Step5: Analyze \( f(x)=3x + 5x^{3} \)

• Calculate \( f(2) \): \( f(2)=3(2)+5(2)^{3}=6 + 40 = 46 \)
• Calculate \( f(-1) \): \( f(-1)=3(-1)+5(-1)^{3}=-3-5=-8 \)
• \( f(2)-f(-1)=46-(-8)=54>0 \). So average rate of change is positive.

Answer:

The functions with a negative average rate of change on the interval from \( x=-1 \) to \( x = 2 \) are:

• \( f(x)=x^{2}-3x - 5 \)
• \( f(x)=3x^{2}-5x \)