Q. 32 - 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.

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

Step-by-Step Solution

Answer

The critical points are $x=\frac{\ln \left(\ln \left(2\right)\right)-\ln \left(\ln \left(3\right)\right)}{\ln \left(3\right)-\ln \left(2\right)}$. The graph of the function is shown below .

Step 1. Given information .

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

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 .

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

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

Step 3. Plot the graph .

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

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