Q. 35 - 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 points.

$f\left(x\right)=\cos x$

Step-by-Step Solution

Answer

The critical point is $x=0$ . The graph of the given function is shown below .

Step 1. Given information .

Consider the given function $f\left(x\right)=\cos x$ .

Step 2. Find the critical points .

To find the critical points differentiate the given function .

$$\begin{gathered} f\left(x\right)= \cos x \\ f\text{'}\left(x\right)= -\sin x \end{gathered}$$

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

$$\begin{gathered} -\sin x =0 \\ x= 0 \end{gathered}$$

Therefore the critical point is $x=0$ .

Step 3. Plot the graph .

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

From the graph the function f has both local maximum and minimum because the turning points are on both positive and negative axis .