4.4 Concavity and Points of Inflection
Chapter
Chapter 4
Section
4.4
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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.

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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.

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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

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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}}

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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}

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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

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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

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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}}

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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}

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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

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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}

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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}

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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}

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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}

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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.

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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.

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0.56mins
Q5b

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

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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

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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}}

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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

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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).

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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.

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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.

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1.33mins
Q12

Use the algorithm for curve sketching to sketch the function

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

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11.55mins
Q13

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?

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1.39mins
Q14
Lectures 3 Videos