Question

If \(0\leq x\leq 1\) and \((sin^{-1}x)^3+(cos^{-1}x)^3 = aπ^3\) then

• A. \(a\geq \frac{1}{32}\)
• B. \(a\geq \frac{1}{16}\)
• C. \(a\leq \frac{1}{32}\)
• D. \(a\leq \frac{1}{16}\)

Hint:

Use s + c = pi/2 and find the minimum of the cubic expression.

Answer 1

The Correct Option is A

Step 1: Check the smallest case:
At $x = \frac{1}{\sqrt2}$, $s = c = \frac\pi4$, so $s^3 + c^3 = 2 \times \frac{\pi^3}{64} = \frac{\pi^3}{32}$ and $a = \frac1{32}$.

Step 2: Check the ends:
At $x = 0$, $s = 0$, $c = \frac\pi2$: sum $= \frac{\pi^3}{8}$, so $a = \frac18$.
So $a$ varies from $\frac1{32}$ to $\frac18$, and the true statement is $a \geq \frac1{32}$.

Final Answer:
$a \geq \frac{1}{32}$, option (A). \[ \boxed{a \geq \frac{1}{32}} \]