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

Step-by-step explanation:

It can be written as,

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

Using the differentiation rule for log function,

`(d)/(dx)(\log_ax)=-(1)/(xloga)`

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

`\begin{gathered} (d)/(dx)(F\left(x\right))=(d)/(dx)(-\log_2\left(x\right)) \\ =-(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