MCQ
One maximum point of ${\sin ^p}x{\cos ^q}x$ is
- ✓$x = {\tan ^{ - 1}}\sqrt {(p/q)} $
- B$x = {\tan ^{ - 1}}\sqrt {(q/p)} $
- C$x = {\tan ^{ - 1}}(p/q)$
- D$x = {\tan ^{ - 1}}(q/p)$
$\frac{{dy}}{{dx}} = p{\sin ^{p - 1}}x.\cos x.{\cos ^q}x + q{\cos ^{q - 1}}x.( - \sin x){\sin ^p}x$
$\frac{{dy}}{{dx}} = p{\sin ^{p - 1}}x.{\cos ^{q + 1}}x - q{\cos ^{q - 1}}x.{\sin ^{p + 1}}x$
Put $\frac{{dy}}{{dx}} = 0$, $\therefore {\tan ^2}x = \frac{p}{q}$
$ \Rightarrow $$\tan x = \pm \sqrt {\frac{p}{q}} $
$\therefore $Point of maxima $x = {\tan ^{ - 1}}\sqrt {\frac{p}{q}} $.
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Then, which of the following statements are true?
$I.$ $g$ has exactly two distinct real roots.
$II$. $g$ can have more than two distinct real roots.
$III$. There exists a real number $\alpha$ such that $g(x) \geq \alpha$ for all real $x$