extreme values of sin⁸x+cos⁸x is/are ki
Answers
Therefore the extreme value of sin⁸x + cos⁸x is 1/8.
Given:
Trigonometric function: sin⁸x + cos⁸x.
To Find:
We have to find the extreme value of sin⁸x + cos⁸x.
Solution:
The given question can be solved very easily as shown below.
Assume that,
f(x) = sin⁸x + cos⁸x.
The above equation can be rearranged as follows.
⇒ f(x) = ( sin⁴x )² + ( cos⁴x )²
The above equation is in the form: a² + b² = ( a + b )² - 2ab.
∴ f(x) = ( sin⁴x + cos⁴x )² - 2. sin⁴x. cos⁴x
⇒ f(x) = [ ( sin²x + cos²x )² - 2. sin²x. cos²x ]² - 2. sin⁴x. cos⁴x
Using the trigonometric identity, sin²x + cos²x = 1, the above equation becomes,
⇒ f(x) = [ 1 - 2. 1/4. ( 2. sinx. cosx )²]² - 2. 1/16. ( 2. sinx. cosx )⁴
We know that. Using this, 2. sinx. cosx = sin 2x, the above equation can be written as,
⇒ f(x) = [ 1 - 1/2. sin² 2x ]² - 1/8. sin⁴ 2x
⇒ f(x) = 1 + 1/4. sin⁴ 2x - sin² 2x - 1/8. sin⁴ 2x
On simplifying, we get,
⇒ f(x) = 1 + 1/8. sin⁴ 2x - sin² 2x
When sin 2x is maximum, we get the extreme value for the above equation.
Sin 2x is maximum when x = 45° then sin 2x = 1
⇒ f(45°) = 1 + 1/8 - 1 = 1/8
Hence, the extreme value of sin⁸x + cos⁸x is 1/8
Therefore the extreme value of sin⁸x + cos⁸x is 1/8.
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