Question

The maximum value of the function \[ f(x) = 3\sin^{12}x + 4\cos^{16}x \] is ?

• A. \(4\)
• B. \(5\)
• C. \(6\)
• D. \(7\)

Hint:

For integer powers of sine or cosine, maxima often occur at boundary values (i.e. 0 or 1).
- Quick check: \(\sin^{12}(\theta)\le 1\) and \(\cos^{16}(\theta)\le 1\).

Answer 1

The Correct Option is A

Step 1: Bounds of \(\sin^{12}x\) and \(\cos^{16}x\).
Because \(-1 \le\sin x\le 1\) and \(-1 \le\cos x\le 1\), we have \(0\le \sin^{12}x \le 1\) and \(0\le \cos^{16}x \le 1\). 

Step 2: Identifying maximum of \(3\sin^{12}x + 4\cos^{16}x\).
We suspect the maximum occurs when one of \(\sin x\) or \(\cos x\) is 1 or 0, because powers flatten any partial values. Indeed, test: \[ \sin x=1 \implies \cos x=0 \quad\Rightarrow\quad f(x)=3\cdot 1 +4\cdot 0=3. \] \[ \sin x=0 \implies \cos x=1 \quad\Rightarrow\quad f(x)=3\cdot 0 +4\cdot 1=4. \] No intermediate combination of \(\sin x,\cos x\) in \((0,1)\) would yield a sum exceeding 4, due to the high exponents diminishing partial values significantly. Hence the maximum is \(\boxed{4}\).