Maximum value of sin^2 theta×cos^2theta
Answers
Answer:
Step-by-step explanation:
Maximum value of sin²theta× cos²theta= 1.
Hope you got it. And it helps you.
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.
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