Question

In Exercises \(23-28\) , find (a) the local extrema, (b) the intervals on which the function is increasing, and (c) the intervals on which the function is decreasing. $$g(x)=x^{1 / 3}(x+8)$$

Short answer

The local extrema are a local minimum at \( x = -8 \) and a local maximum at \( x = 0 \). The function is increasing in the interval \( -8 < x < 0\) and decreasing in the intervals \( x < -8 \) and \( x > 0 \).

Step-by-step solution

Find the derivative of the function

For the given function \( g(x) = x^{1/3}(x+8) \), the derivative \( g'(x) \) can be found using the product rule for differentiation, where the derivative of the product of two functions is the derivative of the first times the second function plus the first function times the derivative of the second. The result is \( g'(x) = \frac{1}{3}x^{-2/3}(x+8) + x^{1/3} \).

Find the critical points

The critical points are found by setting the derivative equal to zero and solving for \( x \). So, \( 0 = \frac{1}{3}x^{-2/3}(x+8) + x^{1/3} \). This gives solutions \( x = 0, x = -8 \). These are the potential points for local extrema.

Determine where the function is increasing or decreasing

Taking numbers less than the smallest critical value, between the critical values, and larger than the largest critical value, one can determine where the function is increasing and where it is decreasing. By testing, it is seen that for values \( x < -8 \), the derivative is negative, which means that the function is decreasing. Between \( -8 < x < 0 \), the derivative is positive, thus the function is increasing. For \( x > 0 \), the derivative is negative again, causing the function to decrease.

Identify the local extrema

Based on previous steps, at \( x = -8 \), the function changes from decreasing to increasing, which means a local minimum is present at \( x = -8 \). At \( x = 0 \), the function changes from increasing to decreasing, which indicates a local maximum exists at \( x = 0 \).

Identify the intervals of increase and decrease

Based on the sign of the derivative in different intervals, it can be concluded that the function is increasing in the interval \( -8 < x < 0 \) and decreasing in the intervals \( x < -8 \) and \( x > 0 \).