Critical Numbers and Derivatives

The video tutorial explains how to find critical numbers of a function f(x) = x * e^(-2x^2). It begins by defining critical numbers as points where the derivative is zero or undefined. The tutorial then demonstrates finding the derivative using the product and chain rules, simplifying the expression, and solving for critical numbers. The critical numbers are determined to be x = ±1/2.

What is the function f(x) given in the video?

• x times e to the power of 2x squared

• x times e to the power of negative 2x squared

• x squared times e to the power of x

• e to the power of x times x squared

What is a critical number of a function?

• A point where the function has a minimum value

• A point where the derivative is zero or undefined

• A point where the function is undefined

• A point where the function has a maximum value

Which rules are applied to find the derivative of f(x)?

• Difference rule and chain rule

• Sum rule and product rule

• Product rule and chain rule

• Quotient rule and chain rule

What is the derivative of x with respect to x?

• x

• 0

• 1

• 2x

What is the derivative of e to the power of negative 2x squared with respect to x?

• e to the power of negative 2x squared

• Negative 4x times e to the power of negative 2x squared

• Negative 2x times e to the power of negative 2x squared

• 2x times e to the power of negative 2x squared

What is the simplified form of the derivative f'(x)?

• e to the power of 2x squared times (1 + 4x squared)

• e to the power of negative 2x squared times (1 + 4x squared)

• e to the power of negative 2x squared times (1 - 4x squared)

• e to the power of 2x squared times (1 - 4x squared)

Why can't e to the power of negative 2x squared be zero?

• Because it is always negative

• Because it is undefined

• Because it is a constant

• Because it is always positive

What equation do we solve to find the critical numbers?

• x squared = 1

• 4x squared = 1

• 1 - 4x squared = 0

• 1 + 4x squared = 0

What are the critical numbers of the function?

• 1/4 and -1/4

• 1 and -1

• 1/2 and -1/2

• 2 and -2

What is the value of x when f'(x) is zero?

• x = 1/2 or x = -1/2

• x = -1

• x = 0

• x = 1