4.4 Concavity and Points of Inflection
Chapter
Chapter 4
Section
4.4
Solutions 25 Videos

For the function, state whether the value of the second derivative is positive or negative at each of points A, B, C, and D. 0.22mins
Q1a

For the function, state whether the value of the second derivative is positive or negative at each of points A, B, C, and D. 0.37mins
Q1b

Determine the critical points for each function, and use the second derivative test to decide if the point is a local maximum, a local minimum, or neither.

y =x^3-6x^2-15x + 10

0.51mins
Q2a

Determine the critical points for each function, and use the second derivative test to decide if the point is a local maximum, a local minimum, or neither.

y = \displaystyle{\frac{25}{x^2 + 48}}

2.25mins
Q2b

Determine the critical points for each function, and use the second derivative test to decide if the point is a local maximum, a local minimum, or neither.

s =t+t^{-1}

0.58mins
Q2c

Determine the critical points for each function, and use the second derivative test to decide if the point is a local maximum, a local minimum, or neither.

y = (x - 3)^3 + 8

0.33mins
Q2d

Determine the points of inflection for each function. Then conduct a test to determine the change of sign in the second derivative.

y =x^3-6x^2-15x + 10

0.59mins
Q3a

Determine the points of inflection for each function. Then conduct a test to determine the change of sign in the second derivative.

y = \displaystyle{\frac{25}{x^2 + 48}}

3.44mins
Q3b

Determine the points of inflection for each function. Then conduct a test to determine the change of sign in the second derivative.

s =t+t^{-1}

1.30mins
Q3c

Determine the points of inflection for each function. Then conduct a test to determine the change of sign in the second derivative.

y = (x - 3)^3 + 8

0.58mins
Q3d

Determine the value of the second derivative at the value indicated. State whether the curve lies above or below the tangent at this point.

f(x) =2x^3-10x + 3, \text{ at x =2}

0.33mins
Q4a

Determine the value of the second derivative at the value indicated. State whether the curve lies above or below the tangent at this point.

 y = x^2 -\displaystyle{\frac{1}{x}}, \text{ at x = -1}

1.43mins
Q4b

Determine the value of the second derivative at the value indicated. State whether the curve lies above or below the tangent at this point.

p(w) = \displaystyle{\frac{w}{\sqrt{w^2 + 1}}}, \text{ at w = 3}

2.00mins
Q4c

Determine the value of the second derivative at the value indicated. State whether the curve lies above or below the tangent at this point.

s(t) = \displaystyle{\frac{2t}{t -4}}, \text{ at t = -2}

1.07mins
Q4d

The graphs represents the second derivative, f''(x), of a function f(x): For each of the graphs above, answer the following questions:

i) On which intervals is the graph of f(x) concave up? On which intervals is the graph concave down?

ii) List the x-coordinates of all the points of inflection.

iii) Make a rough sketch of a possible graph of f(x), assuming that f(0) = 2.

1.06mins
Q5a

The graphs represents the second derivative, f''(x), of a function f(x): For each of the graphs above, answer the following questions:

i) On which intervals is the graph of f(x) concave up? On which intervals is the graph concave down?

ii) List the x-coordinates of all the points of inflection.

iii) Make a rough sketch of a possible graph of f(x), assuming that f(0) = 2.

0.56mins
Q5b

Describe how you would use the second derivative to determine a local minimum or maximum.

0.51mins
Q6

For the following function,

i) determine any points of inflection

ii) use the results of part i, along with the revised algorithm, to sketch each function.

f(x) = x^4 + 4x^3

3.07mins
Q8a

For the following function,

i) determine any points of inflection

ii) use the results of part i, along with the revised algorithm, to sketch each function.

g(x) = \displaystyle{\frac{4x^2 -3}{x^3}}

8.02mins
Q8b

Sketch the graph of a function with the following properties:

f'(x) > 0 when x < 2 and when 2 < x < 5

f'(x) < 0 when x > 5

f'(2) =0 when f'(5) = 0

f''(x) < 0 when x < 2 and when 4 < x < 7

f''(x) > 0 when 2 < x < 4 and when x > 7

f(0) = -4

2.13mins
Q9

Find constants a, b, and c such that the function f(x) = ax^3 +bx^2 +c will have a local extremum at (2, 11) and a point of inflection at (1, 5).

3.46mins
Q10

Find the value of the constant b such that the function f(x) = \sqrt{x + 1} + \displaystyle{\frac{b}{x}} has a point of inflection at x =3.

2.18mins
Q11

Show that the graph of f(x) = ax^4 + bx^3 has two points of inflation. Show that the x-coordinate of one of these points lies midway between the x-intercepts.

1.33mins
Q12

Use the algorithm for curve sketching to sketch the function

y = \displaystyle{\frac{x^3 -2x^2 + 4x}{x^2 - 4}}

Find the inflection points, if any exist, for the graph of f(x) = (x -c)^n, for n= 1, 2, 3, and 4. What conclusion can you draw about the value of n and the existence of inflection points on the graph of f?