What is the maximum value of this function

Calculus Level 2

If f ( x ) = ( sin ⁡ ( x ) cos ⁡ ( x ) ) 2 f(x) = (\sin(x)\cos(x))^2 , what is the maximum value of f ( x ) f(x) ?


The answer is 0.25.

Srinivasa Gopal by Srinivasa Gopal , 53, India

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2 solutions

Srinivasa Gopal
Jul 27, 2018

f(x) = (Sin(x) Cos(x))^2 ; Sin(x) Cos(x) = Sin2x/2; so f(x) = ( Sin(2x)/2)^2; The maximum value of Sin(2x) is equal to 1 and hence the maximum value of f(x) = 1/4 or 0.25

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If the expression sin (x) cos (x) admits maximum value, f (x) will also have maximum value. Then, if sin (x) cos (x) = g (x), g '(x) = cos (x) - sin (x). If we define g (x) = 0 to find a point of minimum or maximum, (cos (x)) 2 - (sin (x)) 2 = 0, cos (x) cos (x)) 2) = 0, 2 (cos (x)) 2 - 1 = 0, cos (x) = + - ((2) ^ 1/2) / 2, (pi (1 + 2k)) / 4, where k is integer. We consider k between 0 and 3, since the angles with negative k or greater than 4 define the same values ​​of sin (x) and cos (x). To ensure that this point is a maximum point, we make the second derivative and see if the concavity is negative. g "(x) = -4cosxsenx. If k = 0, sin (x) and cos (x) will be ((2) ^ 1/2) / 2, both negative, so g "is equal to -2, concavity down. If k = 1, only cos (x) is negative, then the concavity is up. If k = 2, both are negative and the concavity is down. If k = 3, only sin (x) is negative, concave upwards. Therefore, for values ​​of x with k = 1 or k = 3, we will guarantee that g (x) will be maximum, with a value equal to 1/2 = 0.5. Then f (x) will have a maximum value of (0.5) ^ 2 = 0.25

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