I have a strictly positive continuous function $f:=[0,1]\to \mathbb{R}^+$ that I need to maximise. $f$ is actually a polynomial with some parameters on the coefficients, arising in an expected value computation with two events, and the nature of my problem of study suggests strongly that the maximum is attained either in 0 or in 1. I want to prove this, and I am looking for some reference or ideas.
The problem
Is there a criterion to know if a strictly positive, continuous function $f$ defined in $[0,1]$ attains its maximum value on either $x=0$ or $x=1$? What if $f$ is a polynomial?
My attempts (on my particular case)
I first thought that my function was convex in $[0,1]$ (simply plotting $f$ for different sets of parameters), in which case this of course holds, but I found some (not very far-fetched) parameters for which the polynomial was not convex in $[0,1]$, but the property still holds.
The roots cannot be accessed because there are none in $[0,1]$ (strict positiveness). And the other, possibly complex roots are not easy to fetch from the general form.
Then I thought it may also be the case that there is a local maximum (below $f(0)$ and $f(1)$), but I haven't seen that yet from the plots. I tried to look for a differential equation verified by my polynomial in order to get a contradiction with the existence local maximum (a fairly used technique), without real success. Actually, my polynomial looks like
$$f(x)=\alpha(x) + {}_2F_1(a,b,c;x)\cdot \beta(x)$$ where $\alpha,\beta$ are polynomials parameterized by $a,b,c$ (constants of my problem) and ${}_2F_1(a,b,c;x)$ is a polynomial, so I tried to find a differential equation for $f$ using Euler's hypergeometric equation: With the change of variables $v(x) = f(x)/\beta(x)$ and if $$E = x(1-x)\frac{d^2}{dx^2} + (c-(a+b+1)x)\frac{d}{dx} - ab$$ is the Euler operator that vanishes the hypergeometric functions, we have $$E(v(x)-\alpha(x))=0$$ but I didn't know if this helps as it looks kind of artificial.