NIMCET 2012 — Mathematics PYQ

1. Differentiate the function:

Let $y=x^x$.

Taking natural logarithm ($\ln$) on both sides:

Now, differentiate with respect to $x$:

Using the product rule:

2. Set the condition for decreasing function:

For $f(x)$ to be decreasing, f'(x) < 0.

Since $x^x$ is always positive for x > 0, we only need to solve for:

3. Solve for $x$:

Taking the exponential on both sides:

Since the function $x^x$ is defined for x > 0 in this context, the interval is:

Conclusion:

The function decreases in the interval $(0,\frac{1}{e})$.

Correct Option:

(c) $(0,\frac{1}{e})$