Problem 67 - 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 \(f ?\) $$ f(x)=x^{2}-4 x $$

Step-by-Step Solution

Answer

The derivative of the function \(f(x)=x^2 - 4x\) is \(f'(x)=2x - 4\). The \(x\)-intercept of this derivative indicates where the original function \(f(x)\) reaches a local extrema, either a maximum or minimum.

Step 1: Differentiate the Function

The function provided is a simple polynomial. Use the power rule, which states that the derivative of \(x^n\) is \(n*x^{n-1}\). Apply this rule to each term in the equation to find the derivative of the function. The derivative of \(x^2\) is \(2x\) , and the derivative of \(4x\) is \(4\). Therefore, the derivative, \(f'(x)\), of the function \(f(x) = x^2 - 4x \) is \(f'(x)=2x - 4\).

Step 2: Graph the Function and its Derivative

Use a graphing tool to plot both the original function \(f(x)=x^2 - 4x\) and its derivative \(f'(x)=2x - 4\) on the same graph. The choice of tool is up to the individual, but it should be capable of clearly representing both functions together.

Step 3: Interpret the x-intercept of the Derivative

The \(x\)-intercept of the derivative is the value of \(x\) at which the slope of the function \(f\) is \(0\). At this point, \(f(x)\) is at a maximum, minimum, or inflection point. In other words, the \(x\)-intercept of the derivative indicates where the function \(f\) reaches a local extremum (maximum or minimum). Examine the graphs of \(f(x)\) and \(f'(x)\) to confirm this.