Area of Triangle, Sine

Question

In any triangle $ABC$, $a$ and $b$ are the sides of the triangle. Given that $S=\dfrac12ab\sin C$ (where $S$ is the area of the triangle), then

A) $S \geq \dfrac{a^2+b^2}{4}$

B) $S \leq \dfrac{a^2+b^2}{4}$

C) $S \geq \dfrac{a^2+b^2-ab}{2}$

D) $S \leq \dfrac{a^2+b^2-ab}{2}$

E) None of these

Which of the above options is true?

Using AM-GM inequality I got the option (B) as an answer. But it seems option (D) is also valid. But according to the original question, only one of the options is correct.

Answer 1

By AM-GM $$S=\frac{1}{2}ab\sin\gamma\leq\frac{1}{2}ab\leq\frac{1}{2}\cdot\frac{a^2+b^2}{2}=\frac{a^2+b^2}{4}\leq\frac{a^2+b^2-ab}{2},$$ where the last inequality it's $(a-b)^2\geq0.$

Id est, B) and D) they are valid.