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 points.

$f\left(x\right)=3x^4+8x^3-18x^2$

Short answer

The critical points are $x=0,x=1,x=-3$.The graph of the function is shown below .

Step-by-step solution

Step 1. Given information .

Consider the given function$f\left(x\right)=3x^4+8x^3-18x^2$.

Step 2. Find the critical points .

To find the critical points differentiate the given function and put it equal to zero .

$f\text{'}\left(x\right)=0$

$f\left(x\right)=3x^4+8x^3-18x^2$

$f\text{'}\left(x\right)=12x^3+24x^2-36x$

Further simplify .

role="math" localid="1648362135156" $$\begin{gathered} 12x^3+24x^2-36x=0 \\ x\left(12x^2+24x-36\right)=0 \\ x=0,12x^2+24x-36=0 \end{gathered}$$

Further simplify .

$$\begin{gathered} 12x^2+24x-36=0 \\ \left(x+3\right)\left(x-1\right)=0 \\ x=-3,x=1 \end{gathered}$$

Therefore the critical points are$x=0,x=-3,x=1$.

Step 3. Plot the graph .

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

From the above graph f has local minimum because the turning point is on negative axis .