What is the maximum value of sin 2x ⋅ cos 2x?

What is the maximum value of sin 2x ⋅ cos 2x? Correct Answer <span class="math-tex">\(\dfrac{1}{2}\)</span>

CONCEPT:

Second Derivative Test: Let f be a function defined on an interval I.

• Calculate f’(x)
• Solve f’(x) = 0 and find the roots of f'(x) = 0. Suppose x = c is the root of f’(x) = 0.
• Calculate f’’(x) and put x = c to get the value of f’’(c).
• If f’’(c) < 0 then x = c is a point of local maxima.
• If f’’(c) > 0 then x = c is a point of local minima.
• If f’’(c) = 0 then we need to use the first derivative test.

Note:

Maxima and Minima on a closed Interval:

Let f(x) be a given function defined on .Let the local minimum value of f(x) be m and let the local maximum value of f(x) be M. Then

Minimum value of f(x) on is the smallest of m, f(a) and f(b)

Maximum value of f(x) on is the greatest of M, f(a) and f(b)

CALCULATION:

Let f(x) = sin 2x ⋅ cos 2x

⇒ f'(x) = 2 cos² 2x - 2 sin² 2x

⇒ f'(x) = 2 ⋅ (cos² 2x - sin² 2x)

As we know that, cos 2x = cos² x - sin² x

⇒ f'(x) = 2cos 4x

If f'(x) = 0 then 2cos 4x = 0 ⇒ x = π/8

⇒ f''(x) = - 8 sin4x

⇒ f''(x) = - 8 < 0

So, x = π/8 is the point of maxima.

So, the maximum value of f(x) = sin 2x ⋅ cos 2x is given by f(π/8) = sin (π/4) ⋅ cos(π/4) = 1/2

Hence, correct option is 1.