Question

The function ${x^x}$ decreases in the interval-
a) $(0,e)$
b) $(0,1)$
c) $(0,\dfrac{1}{e})$
d)None of the above

Answer 1

Hint : We will be using the derivative technique to solve the given problem. The property of a function to be increasing is that the derivative of that function should be strictly greater than zero and to be decreasing it should be strictly less than zero. We will also be using the logarithmic function to find the derivative of the given function as it is a function involving power of a variable.
Formula used:
1)If in an a mathematical equation $a = b$ , then
$a = b \Leftrightarrow \log a = \log b$
2) $\log ({a^b}) = b\log a$
3) $\dfrac{d}{{dx}}\log x = \dfrac{1}{x}$
4) $\log e = 1$

Complete step by step solution:
The given function is ${x^x}$ .
The given function is defined on the positive real line.
Let $y = {x^x}$
Applying logarithmic function on both sides we get:
$\log y = \log {x^x}$
$\Rightarrow \log y = x\log x$
Differentiating the above equation with respect to $x$ we get:
$\dfrac{1}{y}y' = x\left( {\dfrac{1}{x}} \right) + \log x$
$\Rightarrow y' = y(1 + \log x)$
$\Rightarrow y' = {x^x} + {x^x}\log x$
For $y$ to be an decreasing function $y' < 0$
$\Rightarrow y' < 0 \Rightarrow {x^x} + {x^x}\log x < 0$
$\Rightarrow {x^x}(1 + \log x) < 0$
$\Rightarrow y(1 + \log x) < 0$
The given function $y$ is defined only in the positive real axis so its value is always positive.
Thus, the only possibility we have is:
$1 + \log x < 0$
$\Rightarrow \log e + \log x < 0$
$\Rightarrow \log x < \- \log e$ [Taking $\log e$ to the right side of the inequality]
$\Rightarrow \log x < \log \dfrac{1}{e}$ [we have $- \log e = \log {e^{ - 1}} = \log \dfrac{1}{e}$ ]
Using the properties of logarithmic function we can write:
$x < \dfrac{1}{e}$
Since, in the given case $x > 0$ . So we have the inequality:
$0 < x < \dfrac{1}{e}$
Thus, $x \in (0,\dfrac{1}{e})$ .
Therefore, the function decreases in the interval $(0,\dfrac{1}{e})$
So, the correct answer is “Option C”.

Note : The given function ${x^x}$ is defined only for positive numbers so make sure you solve the problem on the basis of this. While finding derivatives of function like these use the logarithmic function, as it makes differentiation easier.