Problem 68 - 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 \(f ?\) $$ f(x)=2+6 x-x^{2} $$

Step-by-Step Solution

Answer

The derivative of the function \(f(x)\) is \(f'(x) = 6 - 2x\). The derivative equals zero at \(x = 3\), indicating that the function \(f(x)\) has its peak at this point.

Step 1: Derive Function f

The derivative of \(f(x)\) is computed using basic differentiation rules (power rule) applied term by term. Hence, \(f'(x) = 0 + 6 - 2x\).

Step 2: Setting the Derivative Equals to Zero

Stationary points occur where the derivative equals to zero. Therefore, set \(f'(x) = 0~~~=>~~~6 - 2x = 0\). Solving for \(x\), we get \(x = 3\).

Step 3: Drawing the Graphs

Plot the functions \(f(x) = 2 + 6x - x^2\), and its derivative \(f'(x) = 6 - 2x\). Observe that the x-intercept of the derivative graph indicates a stationary point (a peak or trough) on the original function graph.

Step 4: Interpret the Graphs

The x-intercept of the derivative graph shows where the original function changes from increasing to decreasing or vice versa. In this case, the derivative \(f'(x)\) cuts the \(x\)-axis at \(x = 3\), meaning \(f(x)\) peaks at \(x = 3\) (as it changes from increasing to decreasing).