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Let $f: [0, 1]\rightarrow \mathbb{R}$ be a fixed continuous function such that $f$ is differentiable on $(0, 1)$ and $f(0) = f(1) = 0$. Then the equation $f(x) = f' (x)$ admits.

1. No solution $x \in (0, 1)$
2. More than one solution $x \in (0, 1)$
3. Exactly one solution $x \in (0, 1)$
4. At least one solution $x \in (0, 1)$
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+1 vote

all the conditions are given to hold the Rolle's theorem requisites,

hence f′(x) will be 0 for atleast 1 , which gives that f(x) = 0 from the given eqn. hence

f(x) has atleast 1 solution in (0,1).
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