We have, f(x)=
a x + a 2 a x \sqrt{a x}+\frac{a^2}{\sqrt{a x}} a x + a x a 2 On differentiating w.r.t.
x x x , we get,
f ′ ( x ) = a ⋅ 1 2 x + a 2 a ⋅ ( − 1 2 x x ) f ′ ( x ) = a 2 ⋅ 1 x − a a 2 x x f ′ ( a ) = a 2 ⋅ 1 a − a a 2 a a = 1 2 − 1 2 = 0 \begin{aligned}& f^{\prime}(x)=\sqrt{a} \cdot \frac{1}{2 \sqrt{x}}+\frac{a^2}{\sqrt{a}} \cdot \left(-\frac{1}{2 x \sqrt{x}}\right)\\
& f^{\prime}(x)=\frac{\sqrt{a}}{2} \cdot \frac{1}{\sqrt{x}}-\frac{a \sqrt{a}}{2 x \sqrt{x}}\\
& f^{\prime}(a)=\frac{\sqrt{a}}{2} \cdot \frac{1}{\sqrt{a}}-\frac{a \sqrt{a}}{2 a \sqrt{a}}=\frac{1}{2}-\frac{1}{2}=0
\end{aligned} f ′ ( x ) = a ⋅ 2 x 1 + a a 2 ⋅ ( − 2 x x 1 ) f ′ ( x ) = 2 a ⋅ x 1 − 2 x x a a f ′ ( a ) = 2 a ⋅ a 1 − 2 a a a a = 2 1 − 2 1 = 0 