Which one of the following is correct in respect of the function, f ( x ) = x 3 sin x ?

Question

Which one of the following is correct in respect of the function, $f (x) = x^{3} \sin x ?$

Smart Achievers · Accepted Answer

(2)

Ans. It has local minimum at $x = 0$

Explanation:

Given, $f (x) = x^{3} \sin x$

$∴$ $f' (x) = x^{3} \cos x + 3 x^{2} \sin x$

Now, $f'' (x)$ $= - x^{3} \sin x + 3 x^{2} \cos x + 3 x^{2} \cos x + 6 x \sin x$

$= - x^{3} \sin x + 6 x^{2} \cos x + 6 x \sin x$

To find critical points, $f' (x) = 0$

$x^{3} \cos x + 3 x^{2} \sin x = 0$

$x^{2} (x \cos x + 3 \sin x) = 0$

$∴$ $x = 0$

At $x = 0,$

$f'' (x) = 0$

$f''' (x)$ $= - 6 x^{2} \sin x - x^{3} \cos x + 12 x \cos x$ $- 6 x^{2} \sin x + 6 \sin x + 6 x \cos x$

$= - 12 x^{2} \sin x - x^{3} \cos x + 18 x \cos x$ $- 6 x^{2} \sin x$

At $a = 0,$

$f''' (x) = 0$

$f'''' (x)$ $= - 24 x \sin x - 12 x^{2} \cos x - 3 x^{2} \cos x$ $+ x^{3} \sin x + 18 \cos x - 18 x \sin x$ $- 12 x \sin x - 6 x^{2} \cos x$

$= - 42 x \sin x - 21 x^{2} \cos x - 30 x \sin x$ $+ x^{3} \sin x + 18 \cos x$

At $x = 0$ , $f'''' (x) = 18 > 0$

$\Rightarrow$ Function is local minimum at $x = 0$ .

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