Let $G(t,x):[0,1]\times [0,1]\to\mathbb R$ be defined as
$$ G(t,x)=\begin{cases} t(1-x)& \text{if $t\leq x\leq 1$ }\\ x(1-t) & \text{if $x\leq t\leq 1$} \end{cases} $$ For a continuous function $f$ on $[0,1]$ define $$I(f)=\int_0^1\int_0^1G(t,x)f(t)f(x)dtdx$$ Which of the following is true?
$1$. $I(f)>0$ if $f$ is not identically zero.
$2.$ $I(f)=0$ for some non-zero $f$.
$3.$ $I(f)<0$ for some non-zero $f$.
$4$. $I(sin(\pi x))=1.$
It’s single option correct question. How to solve it ? $G(t,x)$ is clearly non negative as given, but value of $f$ is not given so value of $f(t)f(x)$ in integrand can be negative? If minimum value of $f$ is positive then I can say that Integral will be positive . One way can be think about this question as the given function $G(t,x)$ is green function corresponding to ODE $$y’’=-f(x), y(0)=y(1)=0$$ Please help . Thank you.