Calculus: First and Second Derivative

Find the second derivative of the function:
f (x) =  2x - 5x⁶

• f ''(x)= 2 - 30x
• f ''(x) =  2-30x⁵
• f ''(x) = -30x⁵
• f ''(x) = -150x⁴

For a function g(x), g''(3)=-8 indicates that g(x) is ____________ at x=3.

• increasing
• decreasing
• concave up
• concave down

For a function g(x), g'(3) = 0 and g''(3) = -7. This indicates that at x = 3, g(x)

• has a rel max

• has a rel min

• is concave up

• is concave down

• can not be determined

For a function m(x), m'(-5) = 0 and m''(-5) = 0. This indicates that at x = -5, m(x)

• has a rel max

• has a rel min

• is concave up

• is concave down

• can not be determined

On what interval(s) is the function
y = x^3 + 6x^2
concave up?

• (-∞,∞)
• (-∞,-2)
• (-2,∞)
• (0,∞)

• x = -1,1

• x = 0,0

• x = 0,-1

• x = 0,1

How many extrema does the graph of f'(x) have?

• 2
• 3
• 4
• 5

Given the graph of f'(x), what is true about x = -2?

• There is a max at

  x = -2 since f '(x) changes from negative to positive

• There is a max at

  x = -2 since f '(x) changes from positive to negative

• There is a min at

  x = -2 since f '(x) changes from negative to positive

• There is a min at

  x = -2 since f '(x) changes from positive to negative

Given the graph of f'(x), determine the open intervals where f(x) is concave up and concave down?

What is the second derivative test used for in calculus optimization?

• It is used to calculate the area under a curve

• It is used to determine the concavity of a function

• It is used to find the slope of a tangent line

• It is used to determine whether a critical point is a maximum, minimum, or saddle point.