Question 1

Given the graph in question 1, at which $x$-value does the function attain its absolute minimum?

Accepted Answer

$x = 0$

Question 2

For the function $f(x) = -3x^4 + 2x^3 - 36x^2 - 8$, which of the following is a critical point?

Accepted Answer

$x = -3$

Question 3

Given $f(x) = \frac{x^2}{3x+6}$, for $x > -2$, which of the following statements is true about its extrema?

Accepted Answer

It has a relative minimum at $x = 0$

Question 4

For $f(x) = 5x - 4x$, $x > 0$, what is the value of $x$ where the function attains its absolute maximum?

Accepted Answer

There is no absolute maximum

Question 5

If $f(x) = 6x \ln x$, what is the $x$-value of its critical point?

Accepted Answer

$x = 1$

Question 6

Given the profit function $P(x) = -0.3x^3 + \frac{27}{2}x^2 - 60x + 100$, where $x$ is the number of units sold (in thousands), what is the production level $x$ that maximizes profit?

Accepted Answer

$x = 15$

Question 7

Which of the following best describes an absolute maximum of a function $f(x)$?

Accepted Answer

A point where $f(x)$ is greater than or equal to all points in the domain

Question 8

Suppose a company’s cost function is $C(x) = 2x^2 - 12x + 30$. What is the production level $x$ that minimizes cost?

Accepted Answer

$x = 3$

Question 9

If $f(x)$ has a critical point at $x = a$ and $f''(a) > 0$, what can you conclude about $f(x)$ at $x = a$?

Accepted Answer

$f(x)$ has a relative minimum at $x = a$

Question 10

A pharmaceutical company models the revenue from a new drug as $R(x) = 100x - 2x^2$, where $x$ is the number of units sold (in thousands). What is the maximum possible revenue?

Accepted Answer

$R = 2500$

Question 11

Which of the following is NOT a necessary condition for a point $x = c$ to be a local extremum of $f(x)$?

Accepted Answer

$f''(c) 
eq 0$

Question 12

A company’s profit function is $P(x) = -2x^2 + 40x - 100$. What is the maximum profit, and at what value of $x$ does it occur?

Accepted Answer

Maximum profit is $-100$ at $x = 10$

Question 13

If $f(x) = x^3 - 6x^2 + 9x + 1$, what is the $x$-coordinate of the relative minimum?

Accepted Answer

$x = 2$

Question 14

A business wants to maximize the area of a rectangular enclosure with 100 meters of fencing. What is the maximum area?

Accepted Answer

$2500$ square meters

Question 15

Given $f(x) = x^4 - 4x^2$, which of the following is a relative maximum?

Accepted Answer

$x = 0$ and $f(0) = 0$

Question 16

If $f(x)$ is continuous on $[a, b]$ and differentiable on $(a, b)$, which theorem guarantees that $f(x)$ attains both an absolute maximum and minimum on $[a, b]$?

Accepted Answer

Extreme Value Theorem

Question 17

A company’s revenue function is $R(x) = 50x - x^2$, and its cost function is $C(x) = 10x + 20$. What is the production level $x$ that maximizes profit?

Accepted Answer

$x = 10$

Question 18

If $f(x) = x^2 - 4x + 3$, what is the minimum value of $f(x)$?

Accepted Answer

$f(2) = -1$

Question 19

A company’s marginal cost is $C'(x) = 6x - 12$. At what production level is the cost minimized?

Accepted Answer

$x = 2$

Question 20

If $f(x)$ has a local maximum at $x = 1$ and a local minimum at $x = 3$, which of the following must be true?

Accepted Answer

$f'(1) = 0$ and $f'(3) = 0$

Extrema and Optimization in Business Calculus

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