Q. 28 - Calculus

Question

Find the critical points of each function f .Then use a graphing utility to determine whether f has a local minimum, a local maximum, or neither at each of these critical point. $f\left(x\right)=x^3+x^2+1$

Step-by-Step Solution

Answer

The critical points are $x=0 , x= \frac{-2}{3}$. The graph of the given function using graphing utility is shown below .

Step 1. Given information .

Consider the given function $f\left(x\right)=x^3+x^2+1$ .

Step 2. Find the critical points .

To find the critical points differentiate the given function .

$$\begin{gathered} f\left(x\right)=x^3+x^2+1 \\ f\text{'}\left(x\right)=3x^2+2x \end{gathered}$$

Put $f\text{'}\left(x\right)=0$ .

$$\begin{gathered} 3x^2+2x=0 \\ x\left(3x+2\right)=0 \\ x=0 , x=\frac{-2}{3} \end{gathered}$$

Therefore the critical points are $x=0 , x=\frac{-2}{3}$.

Step 3. Plot the graph .

By using graphing utility the graph of the given function is shown below .

From the given graph the function f has local maximum because the turning point is on positive axis .