Inverse Functions and Derivatives

The video tutorial explains how to determine the value of f inverse of 5 and the derivative of f inverse at x=5 for the function f(x) = 1/4 x^3 + x + 1. It uses a theorem that relates the function and its inverse, showing how to find these values without explicitly finding the inverse function. The tutorial includes a graphical representation and step-by-step calculations to solve the problem parts A and B.

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

• f(x) = 1/4 x^3 + x + 1

• f(x) = x^3 + 1/4 x + 1

• f(x) = x^3 + x + 1/4

• f(x) = 1/4 x^2 + x + 1

According to the theorem, if f(a) = b, what is f inverse of b?

• f inverse of b = 1/b

• f inverse of b = 1/a

• f inverse of b = a

• f inverse of b = b

What x value makes f(x) equal to 5?

• x = 0

• x = 1

• x = 2

• x = 3

What is the graphical representation of the inverse function?

• A reflection of the original function across y = x

• A translation of the original function

• A rotation of the original function

• A scaling of the original function

What is the slope of the tangent line to the inverse function at x = 5?

• 1/3

• 1/5

• 1/4

• 1/2

What is the derivative f'(x) of the function f(x)?

• f'(x) = 1/4 x^3 + 1

• f'(x) = 3/4 x^3 + 1

• f'(x) = 3/4 x^2 + 1

• f'(x) = 1/4 x^2 + 1

What is the value of f'(2)?

• 3

• 6

• 4

• 5

What is the derivative of f inverse at x = 5?

• 1/2

• 1/5

• 1/3

• 1/4

What is the relationship between the slopes of the tangent lines of f and f inverse?

• They are squares

• They are negatives

• They are equal

• They are reciprocals

What is the final conclusion about the slope of the tangent line to the inverse function?

• It is equal to 1/4

• It is equal to 2

• It is equal to 4

• It is equal to 3