In Exercises 1-4, find the extreme values and where they occur.
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In Exercises 5-10, identify each \( x \) value at which any absolute extreme value occurs. Explain how your answer is consistent with the Extreme Value Theorem.
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13. \( h(x)=\ln (x+1), \quad 0 \leq x \leq 3 \)
14. \( k(x)=e^{-x^{2}},-\infty<x<\infty \)
15. \( f(x)=\sin \left(x+\frac{\pi}{4}\right), 0 \leq x \leq \frac{7 \pi}{4} \)
16. \( g(x)=\sec x,-\frac{\pi}{2}<x<\frac{3 \pi}{2} \)
17. \( f(x)=x^{2 / 5},-3 \leq x<1 \)
18. \( f(x)=x^{3 / 5},-2<x \leq 3 \)
In Exercises 19-30, find the extreme values of the function and where they occur.
19. \( y=2 x^{2}-8 x+9 \)
20. \( y=x^{3}-2 x+4 \)
21. \( y=x^{3}+x^{2}-8 x+5 \)
22. \( y=x^{3}-3 x^{2}+3 x-2 \)
23. \( y=\sqrt{x^{2}-1} \)
24. \( y=\frac{1}{x^{2}-1} \)
25. \( y=\frac{1}{\sqrt{1-x^{2}}} \)
26. \( y=\frac{1}{\sqrt[3]{1-x^{2}}} \)
27. \( y=\sqrt{3+2 x-x^{2}} \)
28. \( y=\frac{3}{2} x^{4}+4 x^{3}-9 x^{2}+10 \)
29. \( y=\frac{x}{x^{2}+1} \)
30. \( y=\frac{x+1}{x^{2}+2 x+2} \)
Group Activity In Exercises 31-34, find the extreme values function on the interval and where they occur.
31. \( f(x)=|x-2|+|x+3|, \quad-5 \leq x \leq 5 \)
32. \( g(x)=|x-1|-|x-5|,-2 \leq x \leq 7 \)
33. \( h(x)=|x+2|-|x-3|,-\infty<x<\infty \)
34. \( k(x)=|x+1|+|x-3|, \quad-\infty<x<\infty \)
In Exercises 35-42, identify the critical points and determine local extreme values. Identify which critical points are not st points.
35. \( y=x^{2 / 3}(x+2) \)
36. \( y=x^{2 / 3}\left(x^{2}-4\right) \)
37. \( y=x \sqrt{4-x^{2}} \)
38. \( y=x^{2} \sqrt{3-x} \)
39. \( y=\left\{\begin{array}{ll}4-2 x, & x \leq 1 \\ x+1, & x>1\end{array}\right. \)
40. \( y=\left\{\begin{array}{ll}3-x, & x<0 \\ 3+2 x-x^{2}, & x \geq 0\end{array}\right. \)
41. \( y=\left\{\begin{array}{ll}-x^{2}-2 x+4, & x \leq 1 \\ -x^{2}+6 x-4, & x>1\end{array}\right. \)
42. \( y=\left\{\begin{array}{ll}-\frac{1}{4} x^{2}-\frac{1}{2} x+\frac{15}{4}, & x \leq 1 \\ x^{3}-6 x^{2}+8 x, & x>1\end{array}\right. \)
In Exercises \( 11-18 \), use analytic methods to find the extreme values of the function on the interval and where they occur. Identify any critical points that are not stationary points.
11. \( f(x)=\frac{1}{x}+\ln x, \quad 0.5 \leq x \leq 4 \)
\( 12 g(x)=e^{-x},-1 \leq x \leq 1 \)
