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

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

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

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}}

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}

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

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

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}}

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}

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

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}

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}

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}

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}

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.

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.

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

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

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}}

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

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).

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.

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.

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?