Question

The function ${x^x}$ decreases in the interval
A) $(0,e)$
B) $(0,1)$
C) $\left( {0,\dfrac{1}{e}} \right)$
D) None of these

Answer 1

Hint: For this type of question to determine the increase and decrease of the function we first differentiate it and then check in which interval the function is increasing or decreasing.
Complete step by step answer:
Let us assume that $f(x) = {x^x}$ which is also equal to $y = {x^x}$
Then before differentiating let us take log on both the sides
$\therefore \log y = x\log x$
Differentiating both the sides we will get it as
$\Rightarrow \dfrac{1}{y} \times \dfrac{{dy}}{{dx}} = x\dfrac{d}{{dx}}\left( {\ln x} \right) + \ln x \times \dfrac{{dx}}{{dx}}$
We have applied the formula for $\dfrac{d}{{dx}}(uv) = u\dfrac{{dv}}{{dx}} + v\dfrac{{du}}{{dx}}$
And also we know that $\dfrac{d}{{dx}}(\ln x) = \dfrac{1}{x}$
Using all this we can get our next steps as

\Rightarrow \dfrac{1}{y} \times \dfrac{{dy}}{{dx}} = x\left( {\dfrac{1}{x}} \right) + \ln x\\\ \Rightarrow \dfrac{{dy}}{{dx}} = y[1 + \ln x]\\\ \Rightarrow f'(x) = {x^x}[1 + \ln x] \end{array}$$ As $$f(x)$$ decreases $$\begin{array}{l} \Rightarrow f'(x) < 0\\\ \Rightarrow {x^x}[1 + \ln x] < 0 \end{array}$$ Now we know that $$\begin{array}{l} {\log _a}a = 1\\\ \therefore {\log _e}e = 1 \end{array}$$ Replacing this in place of 1 for the value of $$f'(x)$$ , we get $$\begin{array}{l} \Rightarrow \dfrac{{dy}}{{dx}} = y[\ln e + \ln x]\\\ \Rightarrow \dfrac{{dy}}{{dx}} = y[\ln ex] \end{array}$$ As it is known that $${\log _a}x + {\log _a}y = {\log _a}xy$$ Now, $$\begin{array}{l} \Rightarrow {x^x}(\ln ex) < 0\\\ \Rightarrow x < 0,x > \dfrac{1}{e}\\\ or\\\ \Rightarrow x > 0,x < \dfrac{1}{e} \end{array}$$ As $$0 < \dfrac{1}{e}$$ Therefore, $$x > 0,x < \dfrac{1}{e}$$ is true $$ \Rightarrow 0 < x < \dfrac{1}{e}$$ Hence function decreases in the interval $$\left( {0,\dfrac{1}{e}} \right).$$ Therefore option C is correct. Note: It must be noted that the natural log of x i.e., $$\ln x$$ is also equal to $${\log _e}x$$ . Also remember that for increasing decreasing functions always use derivatives to determine if the derivative of a function is greater than 0 then the function is increasing and if the derivative of a function is less than 0 then it is decreasing.