Let $f(x) = 14^{\sin^2 x} + 14^{\cos^2 x}$. The number of integral values that $f(x)$ can take is ____.
Step-by-Step Solution
Key Concept: Finding critical points of a composite function and evaluating extrema at critical points and endpoints determines the range.
For $f(t) = 144^t + 144^{1-t}$ with $t = \sin^2 x$ and $0 \le x \le 1$, compute $f'(t) = \log 144(144^t - 144^{1-t})$. The function is decreasing on $[0, \frac{1}{2}]$ and increasing on $[\frac{1}{2}, 1]$, so $f_{min} = f(\frac{1}{2}) = 24$ and $f_{max} = f(0) = f(1) = 145$.
Correct Answer: 22