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Q.
The function $ y=a(1-\cos x),\,a > 0 $ is maximum when $ x $ is equal to:
Jharkhand CECEJharkhand CECE 2002
Solution:
For maximum or minimum, the second derivative of that function is negative or positive. Given that,
$y=a(1-\cos x)$
On differentiating w.r.t. $x$, we get
$\frac{d y}{d x}=a \sin x$
For maxima or minima, put
$\frac{d y}{d x}=0 $
$\Rightarrow a \sin x=0 \Rightarrow x=0, \pi$
On again differentiating, we get
$\frac{d^{2} y}{d x^{2}}=a \cos x$
At $ x=0, \frac{d^{2} y}{d x^{2}}=a>0,$ minima
At $ x=\pi, \frac{d^{2} y}{d x^{2}}=-a < 0,$ maxima
$\therefore $ Given function is maximum at $x=\pi$.