已知函数 $f(x)=x^\alpha, g(x)=x^\beta$, 其中 $x \in[0,+\infty), 0 < a < 1, \beta>1$, 若点 $M\left(\frac{1}{2}, f\left(\frac{1}{2}\right)\right), N\left(\frac{1}{4}, f\left(\frac{1}{4}\right)\right)$, $P\left(\frac{1}{2}, g\left(\frac{1}{2}\right)\right), Q\left(\frac{1}{4}, g\left(\frac{1}{4}\right)\right)$ 满足 $|M P|=|N Q|$, 则
A. $4^\alpha-4^{\beta}=2^{a+\beta}$
B. $4^a+4^{\beta}=2^{a-\beta}$
C. $2^{\alpha}-2^\beta=2^{\alpha+\beta}$
D. $2^\alpha+2^{\beta}=2^{\alpha+\beta}$