MA005: Calculus I

1. Label all of the local maximums and minimums of the function in Fig. 13. Also label all of the critical points.

Fig. 13

In problems 3-13, find all of the critical points and local maximums and minimums of each function.

3. $f(x)=x^{2}+8 x+7$

5. $f(x)=\sin (x)$

7. $f(x)=(x-1)^{2}(x-3)$

9. $f(x)=2 x^{3}-96 x+42$

11. $f(x)=5 x+\cos (2 x+1)$

13. $f(x)=e^{-(x-2)^{2}}$

15. Sketch the graph of a continuous function $\mathrm{f}$ so that

(a) $f(1)=3, f^{\prime}(1)=0$, and the point $(1,3)$ is a relative maximum of $f$.
(b) $\mathrm{f}(2)=1$, $\mathrm{f}^{\prime}(2)=0$, and the point $(2,1)$ is a relative minimum of $\mathrm{f}$.
(c) $\mathrm{f}(3)=5$, $\mathrm{f}$ is not differentiable at $3$, and the point $(3,5)$ is a relative maximum of $\mathrm{f}$.
(d) $\mathrm{f}(4)=7$, $\mathrm{f}$ is not differentiable at $4$, and the point $(4,7)$ is a relative minimum of $\mathrm{f}$.
(e) $f(5)=4, f^{\prime}(5)=0$, and the point $(5,4)$ is not a relative minimum or maximum of $f$.
(f) $\mathrm{f}(6)=3$, $\mathrm{f}$ is not differentiable at $6$, and the point $(6,3)$ is not a relative minimum or maximum of $\mathrm{f}$.

In problems 17-25, find all critical points and local extremes of each function on the given intervals.

17. $f(x)=x^{2}-6 x+5$ on $[-2,5]$.

19. $f(x)=2-x^{3}$ on $[-2,1]$.

21. $f(x)=x^{3}-3 x+5$ on $[-2,1]$.

23. $f(x)=x^{5}-5 x^{4}+5 x^{3}+7$ on $[0,2]$.

25. $\mathrm{f}(\mathrm{x})=\frac{1}{\mathrm{x}^{2}+1}$ on $[1,3]$.

Fig. 15

29. Find the value for $\mathrm{x}$ so the box in Fig. 17 has the largest possible volume? The smallest volume?

Fig. 17

31. Suppose you are working with a polynomial of degree $3$ on a closed interval.
(a) What is the largest number of critical points the function can have on the interval?
(b) What is the smallest number of critical points it can have?
(c) What are the patterns for the most and fewest critical points a polynomial of degree $\mathrm{n}$ on a closed interval can have?

33. Suppose $f(1)=5$ and $f^{\prime}(1)=0$. What can we conclude about the point $(1,5)$ if
(a) $\mathrm{f}^{\prime}(\mathrm{x}) < 0$ for $\mathrm{x} < 1$, and $\mathrm{f}^{\prime}(\mathrm{x}) > 0$ for $\mathrm{x} > 1$?
(b) $\mathrm{f}^{\prime}(\mathrm{x}) < 0$ for $\mathrm{x} < 1$, and $\mathrm{f}^{\prime}(\mathrm{x}) < 0$ for $\mathrm{x} > 1$?
(c) $\mathrm{f}^{\prime}(\mathrm{x}) > 0$ for $\mathrm{x} < 1$, and $\mathrm{f}^{\prime}(\mathrm{x}) < 0$ for $\mathrm{x} > 1$?
(d) $\mathrm{f}^{\prime}(\mathrm{x}) > 0$ for $\mathrm{x} < 1$, and $\mathrm{f}^{\prime}(\mathrm{x}) > 0$ for $\mathrm{x} > 1$?

35. $\mathrm{f}$ is a continuous function, and Fig. 18 shows the graph of $\mathrm{f} '$
(a) Which values of $\mathrm{x}$ are critical points?
(b) At which values of $\mathrm{x}$ is $\mathrm{f}$ a local maximum?
(c) At which values of $\mathrm{x}$ is $\mathrm{f}$ a local minimum?

Fig. 18

37. State the contrapositive form of the Extreme Value Theorem.

39. Imagine the graph of $\mathrm{f}(\mathrm{x})=1-\mathrm{x}$. Does $\mathrm{f}$ have a minimum value for $\mathrm{x}$ in the interval $\mathrm{I}$?
(a) $\mathrm{I}=[0,2]$
(b) $I=[0,2)$
(c) $\mathrm{I}=(0,2]$
(d) $\mathrm{I}=(0,2)$
(e) $\mathrm{I}=(1, \pi]$

41. Imagine the graph of $\mathrm{f}(\mathrm{x})=\mathrm{x}^{2}$. Does $\mathrm{f}$ have a minimum value for $\mathrm{x}$ in the following intervals?
(a) $\mathrm{I}=[-2,3]$
(b) $\mathrm{I}=[-2,3)$
(c) $\mathrm{I}=(-2,3]$
(d) $\mathrm{I}=[-2,1)$
(e) $\mathrm{I}=(-2,1]$

43. Define $S(x)$ to be the slope of the line through the points $(0,0)$ and $(\mathrm{x}$, $\mathrm{f}(\mathrm{x}))$ in Fig. 21 .
(a) At what value of $\mathrm{x}$ is $\mathrm{S}(\mathrm{x})$ minimum?
(b) At what value of $\mathrm{x}$ is $\mathrm{S}(\mathrm{x})$ maximum?

Fig. 21

Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-4.1-Finding-Maximums-and-Minimums.pdf
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