How to show $\sqrt{f(x)} \geq \sqrt{f(1)} +\frac{1}{2}(x-1)$?

Suppose $f$ is a function such that $f(x) > 0$ and $f'(x)$ is continuous at every real number $x$. If $f'(t) \geq \sqrt{f(t)}$ for all $t$, then show that $$\sqrt{f(x)} \geq \sqrt{f(1)} +\frac{1}{2}(x-1)$$ for all $x \geq 1$.

Trial: Here $f'(t) \geq \sqrt{f(t)} \implies f(t) \geq \int \sqrt{f(t)}~ dt$ Then I am unable to solve

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