Teaching Haryana (HPSC) Assistant Professor Mock Test 2025 Mathematical Science Ordinary Differential Equations Green’s Function
Let G : [0, 1] × [0, 1] → ℝ be defined as
\(G(t, x)=\left\{\begin{array}{l} t(1-x) \text { if } t \leq x \leq 1 \\ x(1-t) \text { if } x \leq t \leq 1 \end{array}\right.\).
For a continuous function f on [0, 1], define
I[f] = \(\rm\int_0^1 \int_0^1 G(t, x) f(t) f(x) d t\ d x\).
Which of the following is true?
1
I[f] > 0 if f is not identically zero.
2
There exists non-zero f such that I[f] = 0.
3
There is f such that I[f] < 0.
4
I[sin (πx)] = 1.
5
Question Not Attempted