Q: A function $$f is $$concave up on an interval if:

$$f''(x) \u003e 0$$ \text{ for all } x \text{ in the interval}$

Question 1

Let $$f(x) = 5\sqrt{x} on $$the interval $$[-1, 1]. $$Compute $$f'(x)$$ and determine if the Mean Value Theorem (MVT) applies to $$f on $$this interval.

Accepted Answer

$$f'(x) = \frac{5}{2\sqrt{x}}$$; MVT does not apply because $$f is $$not differentiable at $$x=0.$$

Question 2

Given $$h(x) = x \ln x on$$ $$[1,2]$$, compute $$h'(x).$$

Accepted Answer

$$h'(x) = \ln x + 1$$

Question 3

For $$h(x) = x \ln x on$$ $$[1,2]$$, does the Mean Value Theorem apply? If so, find the $$c$$ guaranteed by the theorem.

Accepted Answer

Yes, MVT applies. $$c = \frac{1}{\ln 2}.$$

Question 4

Let $$f(x) = 2x^3$ + 3x^2$ - 12x. $$Compute $$f'(x). $$Which of the following is correct?

Accepted Answer

$$f'(x) = 6x^2$ + 6x - 12$$

Question 5

For $$f(x) = 2x^3$ + 3x^2$ - 12x$$, on which intervals is $$f$$ increasing?

Accepted Answer

$$(-\infty, -2) \cup (1, \infty)$$

Question 6

Compute $$f''(x)$$ for $$f(x) = 2x^3$ + 3x^2$ - 12x.$$

Accepted Answer

$$f''(x) = 12x + 6$$

Question 7

For $$f(x) = 2x^3$ + 3x^2$ - 12x$$, at which $$x-$$value(s) does $$f$$ have an inflection point?

Accepted Answer

$$x = -0.5$$

Question 8

A homeowner wants to build a rectangular planting bed with area $$200\ 	ext{ft}^2$$ and width $$x ($$where $$1 \u003c x \u003c 25). $$The bed is against the house, so no fence is needed along one side. Which function gives the total length of the fence needed?

Accepted Answer

$$f(x) = 2\frac{200}{x} + x$$

Question 9

For the planting bed problem, what is the minimum length of fence required (rounded to the nearest foot)?

Accepted Answer

$$40\sqrt{3}\ 	ext{ft}$$

Question 10

Which graph (A, B, C, or D) best represents the function $$f(x) = 2x^3$ + 3x^2$ - 12x$$?

Accepted Answer

Graph C

Question 11

Suppose $$f(x) is $$continuous on $$[a, b]$$ and differentiable on $$(a, b). $$According to the Mean Value Theorem, what must be true?

Accepted Answer

There exists $$c \in (a, b)$$ such that $$f'(c) = \frac{f(b) - f(a)}{b - a}.$$

Question 12

A function $$f is $$concave up on an interval if:

Accepted Answer

$$f''(x) \u003e 0$$ 	ext{ for all } x 	ext{ in the interval}$

Question 13

If $$f'(x) = 0$$ and $$f''(x) \u003c 0 at$$ $$x = a$$, what can you conclude about $$f at$$ $$x = a$$?

Accepted Answer

$$f$$ has a local maximum at $$x = a.$$

Question 14

A real-world application: The velocity $$v(t) of a $$car is given by $$v(t) = 3t^2$ - 12t + 9 (in m/s). At $$what time $$t$$ does the car reach its minimum velocity?

Accepted Answer

$$t = 2$$

Question 15

Suppose $$f(x) is $$decreasing on $$(-\infty, 1)$$ and increasing on $$(1, \infty). $$What can you say about $$x=1$$?

Accepted Answer

$$x=1 is a $$local minimum.

Optimization and Graph Analysis in Calculus

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