Function f(x) = x + 6 x 2 x + 3 x is monotonic increasing, if - Vidyadip

MCQ

Function $f(x) = {{\lambda \sin x + 6\cos x} \over {2\sin x + 3\cos x}}$ is monotonic increasing, if

A
$\lambda > 1$
B
$\lambda < 1$
C
$\lambda < 4$
✓
$\lambda > 4$

✓

Answer

Correct option: D.

$\lambda > 4$

d
(d) The function is monotonic increasing, if $f'(x) > 0$

==> $\frac{{(2\sin x + 3\cos x)\,(\lambda \cos x - 6\sin x)}}{{{{(2\sin x + 3\cos x)}^2}}}$

$ - \frac{{(\lambda \sin x + 6\cos x)(2\cos x - 3\sin x)}}{{{{(2\sin x + 3\cos x)}^2}}} > 0$

==> $3\lambda ({\sin ^2}x + {\cos ^2}x) - 12({\sin ^2}x + {\cos ^2}x) > 0$

==> $3\lambda - 12 > 0$ ==> $\lambda > 4.$

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