Question 1

Given the function $$f(x) = 2x^3$ - 15x^2$ + 24x + 19$$ for $$x \geq 0$$, what is the value of $$x$$ that minimizes $$f(x)$$?

Accepted Answer

$$x = 3$$

Question 2

A ball is thrown straight up and its height at time $$t$$ seconds is given by $$f(t) = 4 + 48t - 16t^2$. At what time does the ball reach its maximum height?

Accepted Answer

$$t = 2$$ seconds

Question 3

For the function $$f(x) = 2x^3$ - 15x^2$ + 24x + 19$$, what is the minimum value of $$f(x)$$ for $$x \geq 0$$?

Accepted Answer

$$f(3) = 46$$

Question 4

A rectangular garden is to be built along one side of a house using 40 feet of fencing for the other three sides. What should the width (the two equal sides perpendicular to the house) be to maximize the area?

Accepted Answer

$$10$$ feet

Question 5

A company wants to minimize the cost function $$C(x) = 2x^3$ - 15x^2$ + 24x + 19$$ for $$x \geq 0. $$What is the first step in finding the value of $$x$$ that minimizes the cost?

Accepted Answer

Set $$C'(x) = 0$$ and solve for $$x$$

Question 6

Suppose you want to maximize the area $$A of a $$rectangular garden with one side against a house and $$40$$ feet of fencing for the other three sides. What is the equation relating the length $$L$$ and width $$W of $$the garden?

Accepted Answer

$$2W + L = 40$$

Question 7

For the ball thrown straight up with height $$f(t) = 4 + 48t - 16t^2$, what is the maximum height reached by the ball?

Accepted Answer

$$52$$ feet

Question 8

Which of the following best describes the process of solving an optimization problem in calculus?

Accepted Answer

Find the derivative, set it equal to zero, solve for critical points, and test endpoints if necessary.

Question 9

A company wants to maximize its profit, modeled by $$P(x) = -2x^2$ + 16x + 5. $$What is the value of $$x$$ that maximizes profit?

Accepted Answer

$$x = 4$$

Question 10

If you are asked to minimize a cost function $$C(x) on $$the interval $$[0, 5]$$, and you find a critical point at $$x = 2$$, what should you do next?

Accepted Answer

Evaluate $$C(x) at$$ $$x = 0$$, $$x = 2$$, and $$x = 5$$ and choose the smallest value.

Question 11

A rectangular garden is to be built along one side of a house using 40 feet of fencing for the other three sides. What is the maximum possible area of the garden?

Accepted Answer

$$100$$ square feet

Question 12

Which of the following is a necessary condition for a function $$f(x) to $$have a maximum or minimum at $$x = a$$?

Accepted Answer

$$f'(a) = 0$$

Question 13

A ball is thrown straight up and its height is given by $$f(t) = 4 + 48t - 16t^2$. What is the velocity of the ball when it reaches its maximum height?

Accepted Answer

$$0 ft/s$$

Question 14

If $$f(x)$$ has a critical point at $$x = 3$$ and $$f''(3) \u003e 0$$, what does this indicate about $$f(x) at$$ $$x = 3$$?

Accepted Answer

$$f(x)$$ has a local minimum at $$x = 3$$

Question 15

A company’s revenue is modeled by $$R(x) = 100x - x^2$. What is the maximum revenue, and at what value of x$ does it occur?

Accepted Answer

Maximum revenue is $$2500 at$$ $$x = 50$$

Question 16

Which of the following best describes why endpoints must be checked when solving optimization problems on a closed interval?

Accepted Answer

Because the maximum or minimum could occur at the endpoints.

Optimization Problems in Business Calculus

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