Chapter Test Max and Min
Chapter
Chapter 3
Section
Chapter Test Max and Min
Solutions 18 Videos

On the interval 0 \leq x \leq 3, the function f(x) =x^2 - 8x +16.

A is always increasing

B is always decreasing

C has a local minimum

D is concave down

Q1

The graph of f'(x) is shown. Which of these statements is not true for the graph of f(x)?

A It has one turning point.

B It is concave down for all values of x.

C It is increasing for x <2.

D It is decreasing for all values of x.

Q2

For a certain function, f’(2)=0 and f’(x) >0 for -1 < x < 2. Which statement is not true?

A (2, f(2)) is a critical point.

B (2, f(2)) is a turning point

C (2, f(2)) is a local minimum

D (2, f(2)) is a local maximum

Q3

If f(x) is an odd function and f(a)= 5, then

A f(-a) = 5

B f(-a) = a

C f(-a) = -5

D f(-a) = -a

Q4

For the function \displaystyle f(x) = \frac{-3}{(x - 2)^2}, which statement is not true?

A The graph has no x-intercepts

B The graph is concave down for all x for which f(x) is defined.

C f'(x)>0 when x < 2 and f'(x) < 0 when x > 2

D \displaystyle \lim_{x \to 2} f(x0 = -\infty 

Q5

Copy and complete this statement.

Given \displaystyle f'(x) = x(x -1)^2 , the graph of f(x) has critical points and turning points.

Q6

The graph of f’(x) is shown. Identify the features on the graph of f(x) at each of points A, B, and C. Be as specific as possible.

Q7

Find the absolute extrema for f(x) = x^3-5x^2 + 6x + 2 on the interval 0 \leq x \leq 4.

Q8

Copy the graph of the function f(x) into your notebook. Sketch the first and second derivatives on the same set of axes.

Q9

Consider the function \displaystyle f(x) = 3x^4 -16x^3 + 18x^2 .

a) How will the function behave as x\to \pm \infty ?

b) Find the critical points and classify them using the second derivative test.

c) Find the locations of the points of inflection.

Q10

The cost, in thousands of dollars, to produce x all-terrain vehicles (ATVs) per day is given by the function C(x) = 0.1x^2 + 1.2x + 3.6.

a) Find the a function U(x) to represent the cost per unit to produce the ATVs.

b) How many ATVs should be produced per day to minimize the cost per unit?

Q11

Consider the function \displaystyle y = x^2 + \frac{1}{x^2} .

a) Identify the vertical asymptote.

b) Find and classify the critical points.

c) Identify the intervals of concavity.

d) Sketch the graph.

Q12

The graph shows the derivative, f’(x), of a function f(x).

a )Which is greater?

• i) f('0) or f'(1)
• ii) f(-1) or f(3)
• iii) f(5) or f(10)

b) Sketch a possible graph of f(x).

Q13

The function g(x) = \frac{1}{(x -a)^2} has vertical asymptote x = a. Without graphing, explain how you know how the graph will behave near x = a.

Q14

A hotel chain typically charges $120 per room and rents an average of 40 rooms per night at this rate. They have found that for each$10 reduction in price, they rent an average of 10 more rooms.

a) Find the rate they should be charging to maximize revenue.

b) How does this change if the hotel only has 50 rooms?

Q15

a) For a given perimeter, what shape of rectangle encloses the most area?

b) For a given perimeter, what type of triangle encloses the most area?

c) Which shape would enclose more area for a given perimeter: a pentagon or an octagon? Explain your reasoning.

d) What two-dimensional shape would enclose the maximum area for a given perimeter?

Q16

In a certain region, the number of bushels of corn per acre, B, is given by the function

B(n) = -0.1n^2 + 10n, where n represents the number of seeds, in thousands, planted per acre.

a) What number of seeds per acre yields the maximum number of bushels of corn?

b) If corn sells for $3/bushel and costs$2 for 1000 seeds, find the optimal number of