Question 1

Suppose $$f(x) is a $$twice-differentiable function on an interval $$I. If$$ $$f''(x) \u003e 0$$ for all $$x in$$ $$I$$, which of the following statements is true about the graph of $$f on$$ $$I$$?

Accepted Answer

The graph of $$f is $$concave up on $$I.$$

Question 2

Given $$f(x) = x^3$ + 3x^2$ - 9x - 13$$, find all critical points and classify each as a relative maximum, minimum, or neither using the Second Derivative Test.

Accepted Answer

Critical points at $$x = -3 ($$relative maximum) and $$x = 1 ($$relative minimum).

Question 3

A company models its profit with $$k(x) = 2e^x - e$$^{2x}$$. Find the x$-coordinate of the relative minimum and the corresponding minimum profit.

Accepted Answer

$$x = 	ext{ln}(0.5)$$, $$k(x) = -0.25$$

Question 4

Which of the following is the correct statement of the Second Derivative Test for relative extrema?

Accepted Answer

If $$f'(c) = 0$$ and $$f''(c) \u003e 0$$, then $$f(c) is a $$relative minimum; if $$f''(c) \u003c 0$$, then $$f(c) is a $$relative maximum.

Question 5

For the function $$g(x) = 10x^3$ - 6x^4$, find all points of inflection.

Accepted Answer

$$x = 0$$ and $$x = rac{1}{	ext{sqrt}(2)}$$

Question 6

A pharmaceutical company models the concentration of a drug in the bloodstream by $$h(x) = e^x - e$$^{2x}$$. At what x$-value does the point of inflection occur?

Accepted Answer

$$x = 	ext{ln}(0.25)$$

Question 7

If $$f''(x)$$ changes sign at $$x = a$$, what does this indicate about the graph of $$f at$$ $$x = a$$?

Accepted Answer

$$x = a is a $$point of inflection.

Question 8

Consider the function $$f(x) = (2x-5)^3$ + 1. $$What is the $$x-$$coordinate of the point where $$f is $$not differentiable, and what does this point represent?

Accepted Answer

$$x = 2.5$$, point of inflection

Question 9

A business models its revenue with $$R(x) = 3x^2$ + 6x - 9. $$Find the critical points and determine if they are maxima, minima, or neither.

Accepted Answer

No critical points; the function is a parabola opening upwards.

Question 10

Which of the following best describes the concavity of $$f(x) if$$ $$f''(x) \u003c 0$$ for all $$x in an $$interval $$I$$?

Accepted Answer

The graph of $$f is $$concave down on $$I.$$

Question 11

A company’s cost function is $$C(x) = x^3$ - 6x^2$ + 9x + 2. $$Find the $$x-$$values of all relative extrema and classify each using the Second Derivative Test.

Accepted Answer

$$x = 1 ($$maximum), $$x = 3 ($$minimum)

Question 12

If $$f'(c) = 0$$ and $$f''(c) = 0$$, which test should be used to classify the critical point at $$x = c$$?

Accepted Answer

First Derivative Test

Question 13

For the function $$f(x) = x^4$ - 4x^2$, find all points of inflection.

Accepted Answer

$$x = -	ext{sqrt}(2)$$ and $$x = 	ext{sqrt}(2)$$

Question 14

A manufacturer’s profit function is $$P(x) = -2x^2$ + 8x + 5. $$Find the $$x-$$value of the relative maximum and the maximum profit.

Accepted Answer

$$x = 2$$, $$P(2) = 13$$

Question 15

Which of the following is NOT a necessary condition for a point of inflection at $$x = a$$?

Accepted Answer

$$f'(a) = 0$$

Question 16

A company’s cost function is $$C(x) = x^3$ - 3x^2$ + 2x. At $$what $$x-$$value does the function have a point of inflection?

Accepted Answer

$$x = 2$$

Question 17

If a function $$f$$ has a critical point at $$x = c$$ and $$f''(c) \u003c 0$$, what does this tell you about $$f at$$ $$x = c$$?

Accepted Answer

$$f(c) is a $$relative maximum.

Applications of Differentiation: Using Second Derivatives to Find Maximum and Minimum Values and Sketch Graphs

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