Given : f(x) = 4x^3 - 6x^2 , cos 2a + 3x , , ,sin 2a . , , sin 6a… - Vidyadip

MCQ

Given : $f(x) = 4x^3 - 6x^2 \, cos 2a + 3x \,\,\,sin 2a .\,\, sin 6a + \sqrt {\ell n\,\,\left( {2\,a\,\, - \,\,{a^2}} \right)}$ then :

A
$f(x)$ is not defined at $x = 1/2$
B
$f ‘ (1/2) < 0$
C
$f ‘ (x) $ is not defined at $x = 1/2$
✓
$f ‘ (1/2) > 0$

✓

Answer

Correct option: D.

$f ‘ (1/2) > 0$

d
$2a - a^2 = - (a^2 - 2a) = - ((a - 1)^2 - 1) = 1 - (a - 1)^2,$

hence $f (x)$ can be defined only when $a = 1$.

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

$f '(\frac{1}{2}) = 3 - 6 cos 2 + 3 sin 2 \,\,sin 6 = 3 (1 + sin 2 \,\, sin 6) - 6 cos 2.$

Note that $cos 2 < 0$ and $1 + sin 2 \,\, sin 6 > 0 ==> D$

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