Find the critical points for the function f (x) = x sin−1 x on the interval [−1, 1]. Identify theabsolute maximum and absolute minimum values. State the intervals for where the functions is increasing,decreasing.

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To find the critical points of the function f(x)=xsin−1(x) on the interval [−1,1], we need to find the values of x for which f′(x)=0 or f′(x) is undefined.First, let&#039;s find the derivative of f(x). We can use the product rule and the chain rule:f′(x)=ddx(xsin−1(x))=sin−1(x)+xddx(sin−1(x))=sin−1(x)+x11−x2Now we need to find where f′(x)=0 or f′(x) is undefined. Since sin−1(x) is defined only on the interval [−1,1], we only need to check the endpoints and the interior points of the interval.At x=−1 and x=1, we have f′(x)=sin−1(x), which is −π2 and π2, respectively. Therefore, f′(x) is not equal to zero at either endpoint.To find the critical points in the interior of the interval, we need to solve the equation f′(x)=sin−1(x)+x11−x2=0. We can rewrite this equation assin−1(x)=−x11−x2.Squaring both sides and simplifying, we getx2=1−x2,which gives us x=±12. However, we need to check whether these values are in the interval [−1,1].Therefore, the critical points are x=−12 and x=12.To identify the absolute maximum and absolute minimum values, we need to evaluate f(x) at the critical points and the endpoints of the interval.At the endpoints, we havef(−1)=−π4andf(1)=π4.At the critical points, we havef(−12)=−π42andf(12)=π42.Therefore, the absolute maximum value of f(x) on the interval [−1,1] is π4, which occurs at x=1, and the absolute minimum value is −π42, which occurs at x=−12.To determine the intervals where the function is increasing or decreasing, we need to examine the sign of f′(x) on each interval. Recall that f′(x)=sin−1(x)+x11−x2.On the interval [−1,−12), f′(x) is negative since sin−1(x) is negative and x11−x2 is also negative. Therefore, f(x) is decreasing on this interval.On the interval (−12,12), f′(x) is positive since sin−1(x) is positive and x11−x2 is also positive. Therefore, f(x) is increasing on this interval.On the interval (12,1], f′(x) is negative since sin−1(x) is positive and x11−x2 is negative. Therefore, f(x) is decreasing on this interval.Therefore, the function f(x)=xsin−1(x) is increasing on the interval (−12,12) and decreasing on the intervals [−1,−12) and (12,1].

Find the critical points for the function f

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