isi-entrance 2019 Q4

isi-entrance · India · UGB Differential equations Integral Equations Reducible to DEs

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a twice differentiable function such that $$\frac{1}{2y} \int_{x-y}^{x+y} f(t)\, dt = f(x), \quad \text{for all } x \in \mathbb{R},\ y > 0$$ Show that there exist $a, b \in \mathbb{R}$ such that $f(x) = ax + b$ for all $x \in \mathbb{R}$.

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a twice differentiable function such that
$$\frac{1}{2y} \int_{x-y}^{x+y} f(t)\, dt = f(x), \quad \text{for all } x \in \mathbb{R},\ y > 0$$
Show that there exist $a, b \in \mathbb{R}$ such that $f(x) = ax + b$ for all $x \in \mathbb{R}$.

Paper Questions

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