First Derivative Test


a. Locate the critical points of f.
b. Use the First Derivative Test to locate the local maximum and minimum values.
c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist).


f(x) = x√(4 - x²) on [-2,2]

Verified step by step guidance

1. To find the critical points of the function f(x) = x√(4 - x²), first compute the derivative f'(x). Use the product rule for differentiation, where if u(x) = x and v(x) = √(4 - x²), then f(x) = u(x)v(x). The product rule states that (uv)' = u'v + uv'.
2. Calculate the derivatives: u'(x) = 1 and v'(x) = (1/2)(4 - x²)^(-1/2) * (-2x) = -x/(√(4 - x²)). Substitute these into the product rule to find f'(x).
3. Set f'(x) = 0 to find the critical points. Solve the equation 1 * √(4 - x²) + x * (-x/(√(4 - x²))) = 0. Simplify and solve for x to find the critical points within the interval [-2, 2].
4. Use the First Derivative Test to determine the nature of each critical point. Evaluate the sign of f'(x) on intervals around each critical point to determine if the function is increasing or decreasing, which will help identify local maxima and minima.
5. To find the absolute maximum and minimum values on the interval [-2, 2], evaluate the function f(x) at the critical points and at the endpoints x = -2 and x = 2. Compare these values to determine the absolute extrema.

Key Concepts

Critical Points

Critical points of a function occur where its derivative is either zero or undefined. These points are essential for identifying local extrema, as they represent potential locations where the function's behavior changes. To find critical points, one must first compute the derivative of the function and solve for values of x that satisfy the derivative equation.

First Derivative Test

The First Derivative Test is a method used to determine whether a critical point is a local maximum, local minimum, or neither. By analyzing the sign of the derivative before and after the critical point, one can conclude if the function is increasing or decreasing. If the derivative changes from positive to negative, the critical point is a local maximum; if it changes from negative to positive, it is a local minimum.

Absolute Maximum and Minimum

The absolute maximum and minimum values of a function on a closed interval are the highest and lowest values the function attains within that interval. To find these values, one must evaluate the function at its critical points and at the endpoints of the interval. The largest and smallest of these values will determine the absolute extrema, which are crucial for understanding the overall behavior of the function on the specified interval.