Question

The ordinate of all points on the curve $y=\frac{1}{2{\sin }^{2}x+3{\cos }^{2}x}$ where the tangent is horizontal, may be

• A. always equal to $1/2$
• B. always equal to $1/3$
• C. $1/2$ or $1/3$ according as $n$ is an even or an odd integer.
• D. $1/2$ or $1/3$ according as $n$ is an odd or an even integer.

Answer 1

Answer: (D)

$y^{\prime }=+\frac{1\cdot 2\sin x\cos x}{{\left(2+{\cos }^{2}x\right)}^2}=0$
$\sin 2x=0$
$x=\frac{n\pi }{2},n\in I$
if $n$ is odd then $y=\frac{1}{2+{\cos }^{2}x}=\frac{1}{2}\left({\cos }^{2}x=0\right)$ if $n$ is even then $y=\frac{1}{2+{\cos }^{2}x}=\frac{1}{3}\left({\cos }^{2}x=1\right)$