Math, asked by vnair603, 9 months ago

Maximum value of sin^2 theta×cos^2theta

Answers

Answered by dunukrish
0

Answer:

Step-by-step explanation:

Maximum value of sin²theta× cos²theta= 1.

Hope you got it. And it helps you.

Answered by ChitranjanMahajan
0

The Maximum value for sin²θcos²θ is 1.

Given

sin²θcos²θ

To Find

Maximum Value of sin²θcos²θ

Solution

To find the maximum value we need to maximise sin²θcos²θ

Let f(θ) = sin²θcos²θ

To maximize f(θ)

  • We need to equate f'(θ)=0
  • Then find values of θ
  • Put the values in f''(θ) to get the correct maximum value

f'(θ)= 2sinθcos³θ - 2sin³θcosθ = 0

Taking 2sinθcosθ common we get

2sinθcosθ(cos²θ-sin²θ) = 0

sin2θcos2θ = 0

From this we get sin2θ = 0 or, cos2θ = 0

Therefore,

θ = 0 or π/2

Taking the second differential we get

f''(θ) = 2cos²2θ - 2sin²2θ

f''(0) = 2 - 0 =2                                                          [sin0 = 0 and cos0 = 1]

f''(π/2) = 0 - 2 = -2                                               [sinπ/2 = 1 and cosπ/2 - 0]

Since the value of f''(θ) is negative for θ= π/2

f(θ) is the maximum for θ= π/2

The maximum value of f(θ)

= sin²π/2 - cos²π/2

= 1 - 0

=1

Hence, the Maximum value for sin²θcos²θ is 1.

#SPJ2

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