91Ó°ÊÓ

You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps. a. Plot the function over the interval to see its general behavior there. b. Find the interior points where \(f^{\prime}=0 .\) (In some exercises, you may have to use the numerical equation solver to approximate a solution.) You may want to plot \(f^{\prime}\) as well. c. Find the interior points where \(f^{\prime}\) does not exist. d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval. e. Find the function's absolute extreme values on the interval and identify where they occur. $$f(x)=x^{3 / 4}-\sin x+\frac{1}{2}, \quad[0,2 \pi]$$

Sketch a smooth connected curve \(y=f(x)\) with \(\begin{aligned}f(-2) &=8 \\\f(0) &=4 \\\f(2) &=0 \\\f^{\prime}(x) &>0 \quad \text { for } \quad|x|>2\end{aligned}\) \(\begin{aligned}&f^{\prime}(2)=f^{\prime}(-2)=0\\\&f^{\prime}(x)<0 \text { for }|x|<2\\\&f^{\prime \prime}(x)<0 \text { for } x<0\\\&f^{\prime \prime}(x)>0 \quad \text { for } \quad x>0\end{aligned}\)

A rocket lifts off the surface of Earth with a constant acceleration of \(20 \mathrm{m} / \mathrm{sec}^{2} .\) How fast will the rocket be going 1 min later?

Find the values of constants \(a, b,\) and \(c\) so that the graph of \(y=a x^{3}+b x^{2}+c x\) has a local maximum at \(x=3,\) local minimum at \(x=-1,\) and inflection point at (1,11).

Find the inflection points (if any) on the graph of the function and the coordinates of the points on the graph where the function has a local maximum or local minimum value. Then graph the function in a region large enough to show all these points simultaneously. Add to your picture the graphs of the function's first and second derivatives. How are the values at which these graphs intersect the \(x\)-axis related to the graph of the function? In what other ways are the graphs of the derivatives related to the graph of the function? $$y=\frac{4}{5} x^{5}+16 x^{2}-25$$