1. Among the four functions $y=2^{x}, y=\log _{2} x, y=x^{2}, y=\cos 2 x$, the number of functions for which $\frac{f\left(x_{1}\right)+f\left(x_{2}\right)}{2} < f\left(\frac{x_{1}+x_{2}}{2}\right)$ always holds when $0 < x_{1} < x_{2} < \frac{\pi}{2}$ is
A. 0
B. 1
C. 2
D. 3
1. If we verify the relationship between function values one by one, it will complicate a simple problem. If we understand the essential properties of the function, the algebraic interpretation of convexity and concavity on an interval is the inequality $f\left(\frac{x_{1}+x_{2}}{2}\right) \leqslant \frac{f\left(x_{1}\right)+f\left(x_{2}\right)}{2}, f\left(\frac{x_{1}+x_{2}}{2}\right) \geqslant \frac{f\left(x_{1}\right)+f\left(x_{2}\right)}{2}$; draw a rough sketch within the interval, pay attention to the understanding of convexity and concavity, $y=2^{x}$ does not satisfy, $y=\log _{2} x$ satisfies, $y=x^{2}$ does not satisfy, $y=\cos 2 x$ does not satisfy, choose $\mathrm{B}$.