Question

For the every value of x the function $f\left(x\right)=\frac{1}{5^x}$is

• A. $Decrea\sin g$
• B. $Increa\sin g$
• C. $Neitherincrea\sin gnordecrea\sin g$
• D. $Increa\sin gforx>0anddecrea\sin gforx<0$

Answer 1

The correct option is A

$Decrea\sin g$

Explanation for correct option:

Step1. Differentiating the function.

Given function,

$$\begin{gathered} f\left(x\right)=\frac{1}{5^x} \\ f\left(x\right)=5^{-x} \end{gathered}$$

Differentiation of $a^x=a^x\log a$

Using the formulae

$$\begin{gathered} f\text{'}\left(x\right)=-5^{-x}\log 5 \\ f\text{'}\left(x\right)=-\log \frac{5}{5^x} \end{gathered}$$

Step2. Finding function for decraesingStep2.

For function to be decreasing $f\text{'}<0$

$$\begin{gathered} f\text{'}\left(x\right)=-\log \frac{5}{5^x} \\ Here,f\text{'}\left(x\right)<0forallx \\ f\left(x\right)isdecrea\sin g \end{gathered}$$

Hence option $\left(A\right)$ is correct answer.