Problem 70 - 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 \(f ?\) $$ f(x)=x^{3}-6 x^{2} $$

Step-by-Step Solution

Answer

The derivative of function \(f(x) = x^{3} - 6x^{2}\) is \(f'(x) = 3x^{2} - 12x\). The x-intercepts of the derivative are \(x = 0\) and \(x = 4\). These indicate the points on the graph of the original function where there is a relative maximum, minimum, or a point of inflection.

Step 1: Calculate the Derivative

The derivative of a function can be obtained by applying the power rule of differentiation, which states that if \(f(x) = x^n\), then \(f'(x) = nx^{n-1}\). Applying this rule, the derivative of \(f(x) = x^{3} - 6x^{2}\) is \(f'(x) = 3x^{2} - 12x.\)

Step 2: Graph the Functions

Use a graphing tool to plot \(f(x) = x^{3} - 6x^{2}\) and its derivative \(f'(x) = 3x^{2} - 12x\). Make sure to plot them in the same viewing window for better comparison.

Step 3: Identify the x-intercept of the Derivative

The x-intercepts of the derivative are the points where the derivative crosses the x-axis. For graph \(f'(x) = 3x^{2} - 12x\), solve the equation for \(x\) when \(f'(x) = 0\). This gives us \(x = 0\) and \(x = 4\). This means that the derivative graph intersects the x-axis at these points.

Step 4: Interpret the x-intercept

The x-intercepts of the derivative function corresponds to the points on the original function \(f(x)\) where the slope of the tangent line is 0. In other words, these are the points where \(f(x)\) has either a relative maximum, minimum, or a point of inflection.