Second Derivative and Stationary Points

The video tutorial explains how to find the relative maximum of a cubic function. It begins by identifying stationary points where the derivative is zero. The second derivative is then used to determine whether these points are relative maxima or minima. The tutorial concludes by identifying the point where the function has a relative maximum.

What is the first step in finding the relative maximum of a cubic function?

• Use the quadratic formula

• Evaluate the function at endpoints

• Calculate the second derivative

• Find the stationary points

What does setting the derivative of a function to zero help us find?

• The function's intercepts

• The function's inflection points

• The function's asymptotes

• The stationary points

What is the derivative of the function given in the video?

• 12x^2 - 3x

• x^3 - 6x^2

• 6x - 12

• 3x^2 - 12x

After factoring the derivative, what are the potential x-values for stationary points?

• x = -1 and x = 2

• x = 2 and x = 5

• x = 0 and x = 4

• x = 1 and x = 3

What is the purpose of using the second derivative in this context?

• To find the function's roots

• To find the function's intercepts

• To determine the concavity at stationary points

• To calculate the function's slope

What is the second derivative of the function in the video?

• 12x^2 - 3x

• 6x - 12

• 3x^2 - 12x

• x^3 - 6x^2

What does a negative second derivative at a stationary point indicate?

• A relative maximum

• A constant slope

• A point of inflection

• A relative minimum

At x = 0, what is the value of the second derivative?

• 0

• 24

• -12

• 12

What does a positive second derivative at a stationary point indicate?

• A relative minimum

• A point of inflection

• A constant slope

• A relative maximum

At x = 4, what is the value of the second derivative?

• -12

• 24

• 0

• 12