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.

f (x) = (x − 1.7) (x + 3)

Short answer

The critical point is $x=\frac{13}{20}$.The graph of the given function using graphing utility is shown below .

Step-by-step solution

Step 1. Given information .

Consider the given function f(x) = (x − 1.7) (x + 3) .

Step 2. Find the critical points .

To find the critical points differentiate the given function .

$$\begin{gathered} f\left(x\right)=\left(x-1·7\right)\left(x+3\right) \\ =x^2+3x-1·7x-5·1 \\ =x^2-1·3x-5·1 \\ f\text{'}\left(x\right)=2x-1·3 \end{gathered}$$

Further simplify .

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

$$\begin{gathered} 2x-1·3=0 \\ 2x=1·3 \\ x=\frac{1·3}{2} \\ x=\frac{13}{20} \end{gathered}$$

Therefore the critical point is$\frac{13}{20}$.

Step 3. Plot the graph .

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

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