Question 1

Given the polynomial function $$f(x) = x^4$ - 4x^2$ + 3$$, which of the following intervals contains all real zeros of $$f(x)$$?

Accepted Answer

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

Question 2

A chemical reaction rate $$R(t) is $$modeled by the polynomial $$R(t) = -2t^3$ + 6t^2$ + 4t$$, where $$t is $$time in seconds. At what value of $$t$$ does the reaction rate reach a local maximum?

Accepted Answer

$$t = 2$$

Question 3

Which of the following statements about the polynomial $$g(x) = x^3$ - 3x^2$ + 2x is $$true?

Accepted Answer

The local minimum occurs at $$x = 2$$

Question 4

A company models its profit $$P(x) in $$thousands of dollars by $$P(x) = -x^2$ + 6x - 5$$, where $$x is $$the number of units sold (in thousands). What is the maximum profit, and at how many units sold does it occur?

Accepted Answer

Maximum profit is $$13 at$$ $$x = 3$$

Question 5

Given the graph of $$f(x)$$ with domain $$-6 \u003c x \u003c 6$$, which of the following is true about the absolute minimum?

Accepted Answer

It occurs at a local minimum inside the interval

Question 6

If $$h(x) is a $$polynomial with domain $$-5 \u003c x \u003c 0$$ and has a local maximum at $$x = -2$$, which of the following must be true?

Accepted Answer

$$h'(-2) = 0$$ and $$h''(-2) \u003c 0$$

Question 7

A population model $$P(t) = t^3$ - 6t^2$ + 9t + 5$$ describes the population of a species over time $$t. At $$what time does the population reach its absolute minimum within the interval $$0 \leq t \leq 5$$?

Accepted Answer

$$t = 3$$

Question 8

Which interval represents the domain of the polynomial $$f(x) = 2x^5$ - x^3$ + 4x - 7$$?

Accepted Answer

$$(-\infty, \infty)$$

Question 9

A medication's effectiveness $$E(x) is $$modeled by $$E(x) = -x^2$ + 8x - 15$$, where $$x is $$the dosage in mg. What is the dosage that maximizes effectiveness?

Accepted Answer

$$x = 4$$

Question 10

Given $$f(x) = x^3$ - 6x^2$ + 9x$$, which of the following is the interval where $$f(x) is $$increasing?

Accepted Answer

$$(0, 3)$$

Question 11

A polynomial $$p(x)$$ has a local minimum at $$x = 1$$ and a local maximum at $$x = -2. $$Which of the following could be the sign of $$p'(x)$$ near $$x = 1$$?

Accepted Answer

Negative to positive as $$x$$ increases through $$1$$

Question 12

If the graph of $$g(x)$$ has an absolute maximum at $$x = -3$$ and an absolute minimum at $$x = 0 on $$the interval $$-5 \u003c x \u003c 0$$, which of the following is true?

Accepted Answer

$$g(-3) \u003e g(x)$$ for all $$x in$$ $$(-5, 0)$$

Question 13

A polynomial $$f(x)$$ models the height of a projectile: $$f(x) = -x^2$ + 10x + 5. $$What is the maximum height reached by the projectile?

Accepted Answer

$$f(5) = 30$$

Question 14

Which of the following intervals describes where the function $$h(x) = x^4$ - 8x^2$ + 16 is $$decreasing?

Accepted Answer

$$(-2, 0)$$ and $$(0, 2)$$

Question 15

A business models its revenue $$R(x) by$$ $$R(x) = x^3$ - 6x^2$ + 9x$$, where $$x is $$the number of products sold. What is the interval of $$x$$ where revenue is maximized?

Accepted Answer

$$x = 3$$

Question 16

Given the polynomial $$f(x) = x^4$ - 4x^2$ + 3$$, which of the following is the correct interval notation for the range of $$f(x)$$?

Accepted Answer

$$[0, \infty)$$

Question 17

A temperature model $$T(x) = -x^2$ + 4x + 1$$ predicts the temperature in degrees Celsius at hour $$x. $$What is the highest temperature predicted, and at what hour does it occur?

Accepted Answer

Highest temperature is $$5 at$$ $$x = 2$$

Question 18

Which of the following intervals contains all $$x$$ for which the polynomial $$p(x) = x^3$ - 3x^2$ + 2x is $$negative?

Accepted Answer

$$(1, 2)$$

Polynomial Functions: Graphs, Extrema, and Interval Notation (Precalculus Review)

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