Proving Fundamental Inequalities in Real Numbers

Prove the following inequalities for all real numbers $a$ and $b$:
a) $a^2 + b^2 \geq 2ab$
b) $\frac{c^2 a^2}{2} + \frac{b^2}{2c^2} \geq ab$ (where $c$ is any non-zero real number)
c) $\frac{a^2 + b^2}{2} \geq \left( \frac{a + b}{2} \right)^2$
d) $\frac{a + b}{2} \geq \frac{2}{\frac{1}{a} + \frac{1}{b}}$ (for $a, b > 0$)