Trigonometric Ratios and Identities
Maximum and Minimum Values
AIEEE 2011
Grade 11
Question:
If $A = \sin^2 x + \cos^4 x$, then for all real $x$:
(1) $1 \leq A \leq 2$
(2) $\frac{3}{4} \leq A \leq 1$
(3) $\frac{13}{16} \leq A \leq 1$
(4) $\frac{3}{4} \leq A \leq \frac{13}{16}$
Step-by-Step Solution
Key Concept: Substitute $t = \sin^2 x \in [0, 1]$. Write $A = t^2 - t + 1$ and find the minimum (vertex) and the values at endpoints $t = 0, 1$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (2)