Question

The function xalogos x is increasing. O A. True ОО O B. False

Answer 1

Given:

$F(x)=\log_{0.5}x$

To check: The function is increasing or not

Explanation:

It can be written as,

$\begin{gathered} F(x)=\log_{\frac{1}{2}}(x) \\ F(x)=-\log_2(x)............(1) \end{gathered}$

Using the differentiation rule for log function,

$\frac{d}{dx}(\log_ax)=-\frac{1}{xloga}$

So, the differentiating the function (1) we get,

$\begin{gathered} \frac{d}{dx}(F\left(x\right))=\frac{d}{dx}(-\log_2\left(x\right)) \\ =-\frac{1}{xlog2} \\ <0 \end{gathered}$

We know that,

If the first derivative of the function,

$f^{\prime}\left(x\right)<0$

Then the function is decreasing.

Using this, we conclude that,

The given function is decreasing function.

Final answer: B. False