If $f$ is uniformly continuous on $[a,\infty)$,and $f’$ has definition on $[a,\infty)$,can one deduce that $f’$ is bounded?

If $ f$ is uniformly continuous on $ [a,\infty)$ ,and $ f’$ has definition on $ [a,\infty)$ ,can one deduce that $ f’$ is bounded on $ [a,\infty)$ ?

I know some functions like $ \sqrt{x}$ which has definition on the open set $ (0,\infty)$ , but which derivative is not bounded, so I wish the boundedness will be hold on a closed set $ [a,\infty)$

finding the extreme values of $f$.

If $ f(x,y) = x ^2 + xy + y^2 – 4 \ln x – 10 \ln y,$ and I found $ f_{x}^{‘} = 2x + y – 4/x$ which means, $ $ 2x^2 + xy = 4………….1.$ $ and also I found $ f_{y}^{‘} = 2y + x – 10/y$ which means, $ $ 2y^2 + xy = 10 ……> 2.$ $

then I subtracted 1. from 2.and I got $ $ y^2 – x^2 = 3,$ $

But then I do not know how to complete, can anyone help me please?