Q. 12 - Precalculus Enhanced with Graphing Utilities

Question

$f\left(x\right)=x^3-3x+5$.

(a) Using a graphing utility, graph $f$ and approximate the zero(s) of $f$.

(b) Using a graphing utility, approximate the local maxima and local minima.

(c) Determine the intervals on which $f$ is increasing.

Step-by-Step Solution

Answer

(a) The graph of the function $f\left(x\right)=x^3-3x+5$ is,

The given function has one real zero and two imaginary zeros and only the real zero is marked in the graph.

(b) The local maxima is, $\left(-1,7\right)$ and the local minima is, $\left(1,3\right)$.

(c) The function can be increasing in the interval $(-\infty ,-1]$ and $[1,\infty )$.

Part a Step 1. Given Information

$f\left(x\right)=x^3-3x+5$.

Let us sketch a graph for the given function.

Part a Step 2 . Determine the holes for the given function.

The given function has one real zero and two imaginary zeros. The real zero is marked in the graph and the imaginary zeros cannot be marked since the imaginary points cannot be marked in the rectangular cartesian coordinate plane.

Part b Step 1 . Determine the local minima and the local maxima.

From the graph, we can see that, the given function has local minima and the local maxima.

The local minima is at $\left(1,3\right)$ and the local maxima is at, $\left(-1,7\right)$.

Part c Step 1 . Determine the intervals on which f is increasing.

From the graph, the function is increasing in the range $(-\infty ,-1]$ and decreasing in the range $\left(-1,1\right)$ and again increasing in the range $[1,\infty )$.