Let f : ℝ → ℝ and g : ℝ → ℝ be functions satisfying f ( x + y ) = f ( x ) + f ( y ) + f (…
Mathematics · JEE Advanced · NTA Exams — Limit, Continuity and Differentiability
Let f : ℝ → ℝ and g : ℝ → ℝ be functions satisfying
f(x + y) = f(x) + f(y) + f(x)f(y) and f(x) = xg(x) for all x, y ∈ℝ. If \(\lim _{x \rightarrow 0} g(x)=1,\) then which of the following statements is/are TRUE?
f(x + y) = f(x) + f(y) + f(x)f(y) and f(x) = xg(x) for all x, y ∈ℝ. If \(\lim _{x \rightarrow 0} g(x)=1,\) then which of the following statements is/are TRUE?
- f is differentiable at every x
- If g(0)=1, then g is differentiable at every x
- The derivative f
- The derivative f
Answer
(D) The derivative f ′ (0) is equal to 1
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