Problem 69 - College Algebra and Calculus: An Applied Approach

Question

Find the derivative of the given function \(f\). Then use a graphing utility to graph \(f\) and its derivative in the same viewing window. What does the \(x\) -intercept of the derivative indicate about the graph of \(f ?\) $$ f(x)=x^{3}-3 x $$

Step-by-Step Solution

Answer

The derivative of \(f(x) = x^{3} - 3x\) is \(f'(x) = 3x^{2} - 3\). The x-intercepts of the derivative are \(x = 1\) and \(x = -1\), indicating the points of inflection in the original function.

Step 1: Find the Derivative of Function

The derivative of \(f(x) = x^{3} - 3x\) can be found by using the power rule, where the derivative of \(x^n\) is \(nx^{n-1}\), and the constant rule, where the derivative of a constant times a variable is the constant. Therefore, \(f'(x) = 3x^{2} - 3\)

Step 2: Find the x-Intercept of Derivative

The x-intercept of the derivative is the the point where \(f'(x) = 0\). Solving \(3x^{2} - 3 = 0\) (i.e. \(x^{2} - 1 = 0\)) gives \(x = 1\) and \(x = -1\). These x-values are the x-intercepts of the derivative.

Step 3: Interpret the Meaning of x-intercept of Derivative

For \(f'(1) = 0\) and \(f'(-1) = 0\), these points are where the original function \(f(x) = x^{3} - 3x\) has a slope of zero, indicating a local minimum or maximum, or a point of inflection. If one observes the graph of the function, it should be apparent that at \(x = 1\) and \(x = -1\), the function changes from decreasing to increasing or vice versa, indicating these are points of inflection.