analyze the function for domain, range, continuity, increasing or decreasing behavior, symmetry…

analyze the function for domain, range, continuity, increasing or decreasing behavior, symmetry, boundedness, extrema, asymptotes, and end behavior. f(x)=sin^(-1)x the function is bounded. identify any local maxima. select the correct choice and, if necessary, fill in the answer box to complete your choice. a. there is a maximum at x = 1. (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.) b. there are no local maxima. identify any local minima. select the correct choice and, if necessary, fill in the answer box to complete your choice. a. there is a minimum at x = . (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.) b. there are no local minima.

analyze the function for domain, range, continuity, increasing or decreasing behavior, symmetry, boundedness, extrema, asymptotes, and end behavior. f(x)=sin^(-1)x the function is bounded. identify any local maxima. select the correct choice and, if necessary, fill in the answer box to complete your choice. a. there is a maximum at x = 1. (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.) b. there are no local maxima. identify any local minima. select the correct choice and, if necessary, fill in the answer box to complete your choice. a. there is a minimum at x = . (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.) b. there are no local minima.

Answer

Answer:

Identify local maxima:

A. There is a maximum at $x = 1$

Identify local minima:

B. There are no local minima

Explanation:

Step1: Recall domain and properties of inverse - sine function

The domain of $y = \sin^{-1}x$ is $[-1,1]$ and the range is $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$. The derivative of $y=\sin^{-1}x$ is $y'=\frac{1}{\sqrt{1 - x^{2}}}$, which is positive for $x\in(- 1,1)$.

Step2: Analyze local maxima

Since the function $y = \sin^{-1}x$ is increasing on its domain $[-1,1]$, the maximum value occurs at the right - hand end of the domain. When $x = 1$, $y=\sin^{-1}(1)=\frac{\pi}{2}$. So there is a maximum at $x = 1$.

Step3: Analyze local minima

Since the function is increasing on $[-1,1]$, the minimum value occurs at the left - hand end of the domain. But we are looking for local minima. A local minimum occurs where the function changes from decreasing to increasing. Since the function is always increasing on its domain, there are no local minima.