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


ƒ(x) = x³ -4a²x

Verified step by step guidance

1. To find the critical points of the function \( f(x) = x^3 - 4a^2x \), we first need to find its derivative. The derivative, \( f'(x) \), will help us identify where the slope of the tangent to the curve is zero or undefined.
2. Differentiate the function \( f(x) = x^3 - 4a^2x \) with respect to \( x \). Using the power rule, the derivative is \( f'(x) = 3x^2 - 4a^2 \).
3. Set the derivative \( f'(x) = 3x^2 - 4a^2 \) equal to zero to find the critical points. This gives the equation \( 3x^2 - 4a^2 = 0 \).
4. Solve the equation \( 3x^2 - 4a^2 = 0 \) for \( x \). This can be done by adding \( 4a^2 \) to both sides and then dividing by 3, resulting in \( x^2 = \frac{4a^2}{3} \).
5. Take the square root of both sides to solve for \( x \). Remember to consider both the positive and negative roots, so \( x = \pm \sqrt{\frac{4a^2}{3}} \). These are 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 identifying local maxima, minima, and points of inflection. To find critical points, one typically takes the derivative of the function and solves for the values of x that satisfy the condition.

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, representing the slope of the tangent line to the curve at any given point. For polynomial functions, the derivative can be calculated using power rules, which simplify the process of finding critical points.

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. The function given, ƒ(x) = x³ - 4a²x, is a cubic polynomial. Understanding the behavior of polynomial functions, including their derivatives, is crucial for analyzing their critical points and overall shape.