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. $$h(x)=\frac{-x}{x^{2}+4}$$

Short answer

The function \(h(x)=\frac{-x}{x^{2}+4}\) has a local minimum at \(x=0\). The function is increasing in the interval \((-∞,0)\) and decreasing in the interval \((0,∞)\).

Step-by-step solution

Compute The Derivative

Given the function \(h(x)=-\frac{x}{{x^2+4}}\). Compute the derivative \(h'(x)\) using the quotient rule which states that \((\frac{f}{g})' = \frac{f'g - fg'}{g^2}\) where \(f(x)=x\) and \(g(x)={x^2+4}\).

Determine The Critical Points

Set the derivative equal to zero and solve for \(x\) to find the possible locations of local extrema. These are the so-called critical points of the function.

Apply The First Derivative Test

Use the first derivative test to classify each critical point of the function. This involves selecting a test point in each interval determined by the critical points and determining whether the function is increasing or decreasing in that interval.

Determine Local Extrema and Intervals of Increase and Decrease

From the sign of the derivative in each interval, establish where the function is increasing and decreasing. According to the first derivative test, a change from positive to negative at a critical point signifies a local maximum, while a change from negative to positive signifies a local minimum.