The function S(x) = _0^x ( t^2 2 ) ,dt has two critical points in… - Vidyadip

MCQ

The function $S(x) =\int\limits_0^x {\sin \left( {\frac{{\pi {t^2}}}{2}} \right)\,dt} $ has two critical points in the interval $[1, 2.4]$. One of the critical points is a local minimum and the other is a local maximum. The local minimum occurs at $x =$

A
$1$
B
$\sqrt 2 $
✓
$2$
D
$\frac{\pi}{2}$

✓

Answer

Correct option: C.

$2$

c
$S(x) = \int\limits_0^x {\sin \left( {\frac{{\pi {t^2}}}{2}} \right)\,dt} $ ;

$S ' (x) = \sin \left( {\frac{{\pi {x^2}}}{2}} \right) = 0$

$\frac{{\pi {x^2}}}{2} = n\pi$ ==> $x^2 = 2n (1 \le x^2 \le 5.76$ as is given)

hence $n = 1$ or $2$

$x=\sqrt 2 $ or $ x = 2$ ;

$S''(x) = cos\left( {\frac{{\pi {x^2}}}{2}} \right). \pi x$

$S''(\sqrt 2) < 0$ and $S''(2) > 0 $

==> minima at $x = 2$

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