Extremum of a sum of polynomial and square root of polynomial

Question

Let $f(x)$ be of the form $f(x) = P_1(x)+\sqrt{P_2(x)}$, where $P_1(x)$ is a monomial $P_1(x)=ax+b$ and $P_2(x)$ is a quadratic function $P_2(x)=cx^2+dx+d$, defined on the closed interval $[0,1]$.

Let also $f'(0)\geq0$ and $f'(1) \geq 0$. Is this enough to say that $f(x)$ has no extrema on the interval $[0,1]$, or do I need further investigation?

If there weren't a square root, I would have a quadratic function so the answer would be positive.

But now I'm stuck with this kind of function. Could you help me please?

Answer 1

$f(x)=-x+2+\sqrt{3 x^2+4 x+5}$

has a minimum at $x=0.115$