Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.


ƒ(x) = x √(x-a)

Verified step by step guidance

1. To find the critical points of the function \( f(x) = x \sqrt{x-a} \), we first need to find its derivative. Start by expressing the function in a form that is easier to differentiate: \( f(x) = x (x-a)^{1/2} \).
2. Apply the product rule to differentiate \( f(x) = x (x-a)^{1/2} \). The product rule states that if \( u(x) \) and \( v(x) \) are functions of \( x \), then \( (uv)' = u'v + uv' \). Here, let \( u(x) = x \) and \( v(x) = (x-a)^{1/2} \).
3. Differentiate \( u(x) = x \) to get \( u'(x) = 1 \). Differentiate \( v(x) = (x-a)^{1/2} \) using the chain rule: \( v'(x) = \frac{1}{2}(x-a)^{-1/2} \cdot 1 = \frac{1}{2\sqrt{x-a}} \).
4. Substitute the derivatives back into the product rule: \( f'(x) = 1 \cdot (x-a)^{1/2} + x \cdot \frac{1}{2\sqrt{x-a}} \). Simplify this expression to find \( f'(x) \).
5. Set \( f'(x) = 0 \) to find the critical points. Solve the equation for \( x \) to determine the values where the derivative is zero or undefined, which will give the critical points of the function.

Key Concepts

Critical Points

Critical points of a function occur where its derivative is either zero or undefined. These points are essential for determining local maxima, minima, and points of inflection. To find critical points, one must first compute the derivative of the function and then solve for the values of x that satisfy the condition of the derivative being zero or undefined.

Derivative

The derivative of a function measures the rate at which the function's value changes as its input changes. It is a fundamental concept in calculus that provides information about the function's slope at any given point. For the function ƒ(x) = x √(x-a), applying the product and chain rules will be necessary to find its derivative.

Function Behavior

Understanding the behavior of a function involves analyzing its continuity, limits, and the nature of its critical points. This includes determining whether critical points correspond to local maxima, minima, or saddle points. By evaluating the second derivative or using the first derivative test, one can gain insights into the function's overall shape and trends.