Critical Points and Derivatives

Vincent explains how to find the critical points of a function by determining where the first derivative is zero or undefined. He uses the function f(x) = x - x^(1/3) and applies the power rule to find the derivative. The video covers identifying critical values by checking where the derivative is zero or does not exist, and solving algebraic equations to find these points. The critical points are x = 0 and x = ±1/(3√3).

What is the function for which we are finding the critical points?

• f(x) = x / cube root of x

• f(x) = x + cube root of x

• f(x) = x - cube root of x

• f(x) = x * cube root of x

Why is the function rewritten as x minus x to the 1/3 power?

• To use the power rule for differentiation

• To simplify the function

• To change the domain of the function

• To make the function continuous

What is the domain of the function f(x) = x - x^(1/3)?

• Only integers

• All negative numbers

• All positive numbers

• All real numbers

What is the first derivative of the function f(x) = x - x^(1/3)?

• 1 + 1/3 * x^(2/3)

• 1 - 1/3 * x^(-2/3)

• 1 - 1/3 * x^(2/3)

• 1 + 1/3 * x^(-2/3)

What condition makes x = 0 a critical point?

• The first derivative is zero

• The first derivative is undefined

• The first derivative is negative

• The first derivative is positive

What is the next step after finding where the derivative is undefined?

• Find where the derivative is continuous

• Find where the derivative equals zero

• Find where the derivative is negative

• Find where the derivative is positive

What algebraic technique is used to solve 1 = 1/(3 * cube root of x^2)?

• Multiplying both sides by 3

• Taking the reciprocal of both sides

• Adding 1 to both sides

• Subtracting 1 from both sides

What is the result of raising both sides of the equation to the third power?

• 1/3 = x^2

• 1/27 = x^2

• 1/3 = x^3

• 1/27 = x^3

What are the final critical values found in the video?

• x = 0, x = ±3√3

• x = 0, x = ±1/√3

• x = 0, x = ±1/3√3

• x = 0, x = ±1/3

What is the relationship between critical values and critical points?

• Critical values are a subset of critical points

• Critical points are a subset of critical values

• They are completely different concepts

• They are interchangeable terms