Chapter 24

Use the second derivative to state whether each curve is concave upward or concave downward at the given value of \(x .\) Check by graphing. $$y=x^{4}+x \text { at } x=1$$

Graph the region bounded by the given curves. \(y=4 / x, y=x, x=6,\) and the \(x\) axis

Use derivatives to find any maximum and minimum points for each function. Distinguish between maximum and minimum points by graphing calculator, by the first-derivative test, the second-derivative test, or the ordinate test. Check by graphing. $$y=x^{3}-7 x^{2}+36$$

Use derivatives to find any maximum and minimum points for each function. Distinguish between maximum and minimum points by graphing calculator, by the first-derivative test, the second-derivative test, or the ordinate test. Check by graphing. $$y=2 x^{2}-x^{4}$$

Use derivatives to find any maximum and minimum points for each function. Distinguish between maximum and minimum points by graphing calculator, by the first-derivative test, the second-derivative test, or the ordinate test. Check by graphing. $$y=3 x^{4}-4 x^{3}-12 x^{2}$$

Use derivatives to find any maximum and minimum points for each function. Distinguish between maximum and minimum points by graphing calculator, by the first-derivative test, the second-derivative test, or the ordinate test. Check by graphing. $$y=2 x^{3}-9 x^{2}+12 x-3$$

Use derivatives to find any maximum and minimum points for each function. Distinguish between maximum and minimum points by graphing calculator, by the first-derivative test, the second-derivative test, or the ordinate test. Check by graphing. $$y=x^{3}+3 x^{2}-9 x+5$$

Use derivatives to find any maximum and minimum points for each function. Distinguish between maximum and minimum points by graphing calculator, by the first-derivative test, the second-derivative test, or the ordinate test. Check by graphing. $$y=(x-2)^{2}(2 x+1)$$

Use the second derivative to find any inflection points for each function. Check by graphing. $$y=2 x^{3}+x^{2}-3$$

Use the second derivative to find any inflection points for each function. Check by graphing. $$y=x^{4}-x^{3}+1$$

Use the second derivative to find any inflection points for each function. Check by graphing. $$y=x^{4}+2 x^{3}+2 x-3$$

Use the second derivative to find any inflection points for each function. Check by graphing. $$y=x^{3}-x$$

Use the second derivative to find any inflection points for each function. Check by graphing. $$y=4 x-3 x^{3}$$

Use the second derivative to find any inflection points for each function. Check by graphing. $$y=5 x^{3}-2 x^{2}+1$$

Use the second derivative to find any inflection points for each function. Check by graphing. $$y=x^{3}-5 x-1$$