Practicing Success
For the function $f(x)=\int\limits_0^x \frac{\sin t}{t} d t$, where $x>0$. At $x=n \pi, f(x)$ attains |
maximum or minimum according as $n$ is odd or even respectively minimum or maximum according as $n$ is odd or even respectively maximum at $x=n \pi$ minimum at $x=n \pi$ |
maximum or minimum according as $n$ is odd or even respectively |
We have, $f(x) =\int\limits_0^x \frac{\sin t}{t} d t$ $\Rightarrow f'(x) =\frac{\sin x}{x}$ and $f''(x)=\frac{x \cos x-\sin x}{x^2}$ For maximum/minimum, we must have $f'(x)=0$ $\Rightarrow \frac{\sin x}{x} =0 \Rightarrow \sin x=0 \Rightarrow x=n \pi, n \in N$ [∵ x > 0] At $x=n \pi$, we have $f''(x)=\frac{n \pi \cos n \pi-\sin n \pi}{n^2 \pi^2}=\frac{(-1)^n}{n \pi}$ $\Rightarrow f''(x) < 0$, if n is an odd natural number and, f''(x) > 0, if n is an even natural number. Hence, f(x) attains local maximum or minimum at $x=n \pi$ according as n is an odd natural number or an even natural number. |