Question

the graph shows a polynomial function f. polynomial function g is defined by ( g(x) = x^2(9 - x) ). compare the maximum values and the end behavior of the functions f and g. compare the maximum values of f and g when ( x > 0 ). the minimum value of ( f(x) ) is (square) which is less than the maximum value of ( g(x) ) which is near (square) (question continued...)

Explanation:

Step1: Expand the function

First, expand \( g(x)=x^{2}(9 - x) \) to \( g(x)=9x^{2}-x^{3} \).

Step2: Find the derivative

To find the maximum, take the derivative of \( g(x) \). The derivative \( g^{\prime}(x) \) is \( g^{\prime}(x)=18x - 3x^{2} \).

Step3: Find critical points

Set \( g^{\prime}(x) = 0 \), so \( 18x-3x^{2}=0 \). Factor out \( 3x \): \( 3x(6 - x)=0 \). The critical points are \( x = 0 \) and \( x = 6 \). Since \( x>0 \), we consider \( x = 6 \).

Step4: Evaluate the function at critical point

Substitute \( x = 6 \) into \( g(x) \): \( g(6)=6^{2}(9 - 6)=36\times3 = 108 \).

Step5: Check end behavior (but for maximum in \( x>0 \), we focus on critical point)

As \( x\to\infty \), \( g(x)=9x^{2}-x^{3}\to-\infty \) (since the cubic term dominates). At \( x = 6 \), we have a local maximum (and since it's the only critical point for \( x>0 \) that gives a peak, it's the maximum).

Answer:

The maximum value of \( g(x) \) when \( x>0 \) is \( 108 \).