Consider the function f(x) = sin (x) in the interval \[{\text{x}} \in \left[ {\frac{\pi }{4},\,\frac{{7\pi }}{4}} \right].\] The number and location(s) of the local minima of this function are A. One, at \[\frac{\pi }{2}\] B. One, at \[\frac{{3\pi }}{2}\] C. Two, at \[\frac{\pi }{2}\] and \[\frac{{3\pi }}{2}\] D. Two, at \[\frac{\pi }{4}\] and \[\frac{{3\pi }}{2}\]

” option2=”One, at \[\frac{{3\pi }}{2}\]” option3=”Two, at \[\frac{\pi }{2}\] and \[\frac{{3\pi }}{2}\]” option4=”Two, at \[\frac{\pi }{4}\] and \[\frac{{3\pi }}{2}\]” correct=”option1″]

The correct answer is $\boxed{\text{A. One, at }\frac{\pi}{2}}$.

The function $f(x) = \sin(x)$ is

a continuous function on the interval $\left[ \frac{\pi}{4}, \frac{7\pi}{4} \right]$. It has a local minimum at $x = \frac{\pi}{2}$, since the derivative $f'(x) = \cos(x)$ is positive on $\left[ 0, \frac{\pi}{2} \right]$ and negative on $\left[ \frac{\pi}{2}, 2\pi \right]$.

The other options are incorrect because they either state that there are two local minima or that the local minima are at different points.