Question

question 5 (1 point)
graph the function:
( h(x) = x^3 + 2x^2 )
then determine whether the function is increasing, decreasing, constant, or undefined over the interval ( (0, infty) ).
( \bigcirc ) a increasing
( \bigcirc ) b decreasing
( \bigcirc ) c constant
( \bigcirc ) d undefined

Explanation:

Step 1: Find the derivative of \( h(x) \)

To determine if a function is increasing or decreasing over an interval, we can use the derivative. The function is \( h(x)=x^{3}+2x^{2} \). Using the power rule (\( \frac{d}{dx}x^{n}=nx^{n - 1} \)), the derivative \( h'(x) \) is:
\( h'(x)=3x^{2}+4x \)

Step 2: Analyze the derivative over \( (0,\infty) \)

For \( x\in(0,\infty) \), \( x>0 \). Let's look at each term of \( h'(x) = 3x^{2}+4x \).

• The term \( 3x^{2} \): Since \( x>0 \), \( x^{2}>0 \), so \( 3x^{2}>0 \).
• The term \( 4x \): Since \( x > 0 \), \( 4x>0 \).

The sum of two positive numbers is positive. So \( h'(x)=3x^{2}+4x>0 \) for all \( x\in(0,\infty) \).

A function is increasing on an interval if its derivative is positive on that interval. So \( h(x) \) is increasing on \( (0,\infty) \).

Answer:

a. Increasing