Question

The function $y=a(1-\cos x),a>0$ is maximum when $x$ is equal to:

• A. $\pi$
• B. $\frac{\pi }{2}$
• C. $-\frac{\pi }{2}$
• D. $-\frac{\pi }{6}$

Answer 1

Answer: (A)

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
$\frac{dy}{dx}=a\sin x$
For maxima or minima, put
$\frac{dy}{dx}=0$
$\Rightarrow a\sin x=0\Rightarrow x=0,\pi$
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 $x=\pi$.