Q. 31 - 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.

$f\left(x\right)=3x-2e^x$

Step-by-Step Solution

Answer

The critical point is $x= In\left(\frac{3}{2}\right)$ . The graph of the given function is shown below .

Step 1. Given information .

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

Step 2. Find the critical points .

The critical points are the points where the function is defined and its derivative is zero or undefined .

Differentiate the given function .

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

Therefore the critical point is $x=In \left(\frac{3}{2}\right)$ .

Step 3. Plot the graph .

The graph of the given function is shown below .

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