The maximum value of sin x . cos x is ______.

Question

KollegeApply · Accepted Answer

The maximum value of sin x . cos x is $\frac{1}{2}$. Explanation: We have f(x) = sin x cos x $\Rightarrow$ f(x) = $\frac{1}{2}$ * 2 sin x cos x = $\frac{1}{2}$ sin 2x f'(x) = $\frac{1}{2}$ * 2 cos 2x $\Rightarrow$ f'(x) = cos 2x Now for local maxima and local minima f'(x) = 0 $\therefore$ cos 2x = 0 2x = ("n" + 1) pi/2, n ∈ I $\Rightarrow$ x = (2"n" + 1) pi/4 $\therefore$ x = pi/4, (3pi)/4 ..... f"(x) = – 2 sin 2x "f''"(x)_(x = pi/4) = -2 sin 2 * pi/4 = - 2 sin pi/2 = – 2 < 0 maxima `"f''"(x)_(x = (3pi)/4) = - 2 sin 2 * (3pi)/4 = -2 sin (3pi)/4` = 2 > 0 minima So f(x) is maximum at x = pi/4 $\therefore$ Maximum value of f(x) = sin pi/4 * cos pi/4 = 1/sqrt(2) * 1/sqrt(2) = 1/sqrt(2).

AChoose Your Answer