Question

If $f(x) = sin^{6}x + cos^{6}x$ then the range of $f(x)$ is

• A. $(1/4, 3/4)$
• B. $[1/4, 1]$
• C. $[1/4, 3/4]$
• D. $[3/4, 1]$

Hint:

$sin^{4}x + cos^{4}x = 1 - \frac{1}{2}sin^{2}2x$ and $sin^{6}x + cos^{6}x = 1 - \frac{3}{4}sin^{2}2x$.

Answer 1

The Correct Option is B

Step 1: Concept
Use the identity $a^{3} + b^{3} = (a+b)(a^{2} - ab + b^{2})$. Here, $(sin^{2}x)^{3} + (cos^{2}x)^{3}$.

Step 2: Meaning
$f(x) = (sin^{2}x + cos^{2}x)(sin^{4}x - sin^{2}x cos^{2}x + cos^{4}x) = 1 - 3 sin^{2}x cos^{2}x$.

Step 3: Analysis
$f(x) = 1 - \frac{3}{4} sin^{2}(2x)$. Since $0 \le sin^{2}(2x) \le 1$.

Step 4: Conclusion
The maximum value is $1 - 0 = 1$. The minimum value is $1 - 3/4 = 1/4$. Thus, the range is $[1/4, 1]$. Final Answer: (B)