Question

Consider the function on the interval \((0,2 \pi)\) For each function, (a) find the open interval(s) on which the function is increasing or decreasing, (b) apply the First Derivative Test to identify all relative extrema, and (c) use a graphing utility to confirm your results. $$ f(x)=\sin ^{2} x+\sin x $$

Short answer

The function \(f(x)=\sin^{2}x+\sin x\) on the interval \((0,2\pi)\) increases on \((0, \frac{\pi}{2})\) and \((\frac{3\pi}{2}, 2\pi)\), and decreases on \((\frac{\pi}{2}, \frac{3\pi}{2})\). The relative maximum occurs at \(\frac{\pi}{2}\) and relative minimum occurs at \(\frac{3\pi}{2}\).

Step-by-step solution

Differentiation

The first step is to find the derivative of the function \(f(x)=\sin ^{2} x+\sin x\). The derivative of \(\sin x\) is \(\cos x\), and using the chain rule, the derivative of \(\sin^2 x\) is \(2\sin x\cos x\). With this, we get \(f'(x)=2\sin x \cos x + \cos x\).

Finding critical points

Next, we find the critical points by setting the first derivative equal to zero: \(0 = 2\sin x \cos x + \cos x\), which simplifies to \(0 = \cos x(2 \sin x + 1)\). From this equation, critical points occur when \(\cos x = 0\) or \((2\sin x + 1) = 0 \), which gives us \(x = \frac{\pi}{2}, \frac{3\pi}{2}, –\frac{1}{2}\). Since \(-\frac{1}{2}\) doesn't fall within the interval (0,2\pi), our critical points are \(x = \frac{\pi}{2}\), and \(\frac{3\pi}{2}\).

First Derivative Test

Now we determine the intervals of increase and decrease. This can be done by taking values in the intervals \((0, \frac{\pi}{2})\), \((\frac{\pi}{2}, \frac{3\pi}{2})\), and \((\frac{3\pi}{2}, 2\pi)\) and evaluating them in the derivative function. After doing so we find that the function increases on the interval \((0, \frac{\pi}{2})\), decreases on \((\frac{\pi}{2}, \frac{3\pi}{2})\), and increases on \((\frac{3\pi}{2}, 2\pi)\). The relative maximum occurs at \((\frac{\pi}{2}, f(\frac{\pi}{2}))\) and relative minimum occurs at \((\frac{3\pi}{2}, f(\frac{3\pi}{2}))\)

Confirm with Graph

Finally, we use a graphing utility to sketch the function within the given interval. The function’s critical points and areas of increase and decrease should align with the calculations.