Minimum value of expression containing $2$ variables $x,y$ 
Finding minimum of $\displaystyle f(a,b)= (a-b)^2+\bigg(\frac{a^2}{20}-\sqrt{(17-b)(b-13)}\bigg)^2,$
  where $a\in\mathbb{R^{+}},b\in(13,17)$ 
  (Without using Geometry)

Try: Using partial derivative with r to $a$ (Treating $b$ as a constant)
$\displaystyle f'(a,b)= 2(a-b)+2\bigg(\frac{a^2}{20}-\sqrt{(17-b)(b-13)}\bigg)\cdot \frac{a}{10}$
Using partial derivative with r to $b$(Treating $a$ as a constant)
$\displaystyle f'(a,b)= -2(a-b)+2\bigg(\frac{a^2}{20}-\sqrt{(17-b)(b-13)}\bigg)\cdot (2b-30)$(Treating $b$ as a constant)
Could some help me to solve it, thanks

As analysis teaches us, take the derivatives with respect to $a$ and $b$ separately, and converge them into a system of equations imposing them to be zero:
$$\begin{cases}
\large \frac{a^3}{100}+a \left(2-\frac{1}{5} \sqrt{-b^2+30 b-221}\right)-2 b = 0\\\\
\large \frac{a^2 (b-15)}{10 \sqrt{-b^2+30 b-221}}-2 a+30 = 0
\end{cases}$$
Now you have to solve them both for $a$ and $b$
One of the solutions (the only real one) is simple and it's
$$a = 10$$
$$b = 15 - \sqrt{2}$$
Substituting those values into the initial equation gives you the minimum value which is
$$f(10, 15 - \sqrt{2}) = \left(\sqrt{2}-5\right)^2+\left(5-\sqrt{\left(2-\sqrt{2}\right) \left(\sqrt{2}+2\right)}\right)^2 = \color{red}{54 - 20\sqrt{2}}$$
As Dr. Sonnhard Gaubner already told you.
