Let f(x) be a real differentiable function such that f(0)=1 and f(x+y)=f(x) f^ (y)+f^ (x) f(y) for all x, y ∈ R . Then ∑_ n=1 ^ 100 _ e f(n) is equal to :

Correct answer: A. 2525 f(x+y)=f(x) · f(y)+f(x) · f(y), x, y ∈ R ....(i) And f(0)=1 ....(ii) Now replace x by zero and y by zero we get aligned & f(0)=f(0) f(0)+f(0) f(0) & 1=f(0)+f(0) & f^ (0)=(1)/(2) ...(iii) aligned Now replace y by zero in equation (i), we get f(x)=(1)/(2) f(x)+f^ (x) or, (1)/(2) f(x)=f^ (x) then f^ (x) f(x) =(1)/(2) hence |f(x)|=(x)/(2)+c Put x=0 , we get c=0 |f(x)|=(x)/(2) Then ∑_ n=1 ^ 100 (f( ))=((1)/(2)+(2)/(2)+(3)/(2)+ +(100)/(2)) =(5050)/(2)=2525

JEE Main 202522 JanuaryMorning Shift

Exam: JEE Main
Year: 2025
Date: 22 January
Shift: Morning Shift
Subject: Mathematics
Chapter: Continuity and Differentiability

JEE Main 2025 · 22 January · Morning Shift · Mathematics · Continuity and Differentiability

Let f(x) be a real differentiable function such that f(0)=1 and...

Let

f(x)

be a real differentiable function such that

f(0)=1

and

f(x+y)=f(x) f^{\prime}(y)+f^{\prime}(x) f(y)

for all

x, y \in \mathbf{R}

. Then

\sum_{\mathrm{n}=1}^{100} \log _{\mathrm{e}} f(\mathrm{n})

is equal to :

Options

A $2525$
B $5220$
C $2384$
D $2406$

Correct answer

A. $2525$

Step-by-step solution

\because f(x+y)=f(x) \cdot f(y)+f(x) \cdot f(y), \forall x, y \in R

....(i)

And f(0)=1

....(ii) Now replace

x

by zero and

y

by zero we get

\begin{aligned} & f(0)=f(0) f(0)+f(0) f(0) \\ & 1=f(0)+f(0) \\ & \therefore \quad f^{\prime}(0)=\frac{1}{2} ...(iii) \end{aligned}

Now replace

y

by zero in equation (i), we get

f(x)=\frac{1}{2} f(x)+f^{\prime}(x)

or,

\frac{1}{2} f(x)=f^{\prime}(x)

then

\frac{f^{\prime}(x)}{f(x)}=\frac{1}{2}

hence

\ln |f(x)|=\frac{x}{2}+c

Put

x=0

, we get

c=0

\therefore \quad \ln |f(x)|=\frac{x}{2}

Then

\sum_{n=1}^{100} \ln (f(\eta))=\left(\frac{1}{2}+\frac{2}{2}+\frac{3}{2}+\ldots+\frac{100}{2}\right)

=\frac{5050}{2}=2525

Practice Continuity and Differentiability on Quantrex Academy →

More from Continuity and Differentiability

Let ℝ denote the set of all real numbers. Let f : ℝ arrow ℝ be an arbitrary function and… JEE Advanced 2026 Consider the function f : (-(π)/(2), (π)/(2)) arrow (-∞, ∞) defined by f(x) = (|x| + |x… JEE Advanced 2026 For a real number α , let [α] denote the greatest integer less than or equal to α . For a… JEE Advanced 2026 For the function f(x) = e^ |x| - |x| , x ∈ ℝ , consider the following statements… JEE Main 2026 Let f(x) = cases (1)/(3), & x ≤ π/2 (b(1- x))/((π-2x)^2), & x > π/2 cases . If f is… JEE Main 2026 Let f(x) = cases x^3 + 8 ; & x Then the number of points, where the function g ∘ f is… JEE Main 2026 Let f(x)= cases e^ x-1 , & x<0 x^2-5x+6, & x ≥ 0 cases and g(x)=f(|x|)+|f(x)| . If the… JEE Main 2026 The number of points, at which the function f(x) = \ 6x, 2 + 3x^2\ + |x - 1|| |x^2… JEE Main 2026 The number of points in the interval [2, 4] , at which the function f(x) = [x^2 - x… JEE Main 2026 Let f(x)= _ θ arrow 0 ( π x-x^ ((2)/(θ)) (x-1) 1+x^ ((2)/(θ)) (x-1) ), x ∈ R . Consider… JEE Main 2026 Let [t] denote the greatest integer less than or equal to t . If the function f(x)=\… JEE Main 2026 If the function f(x)= e^ x (e^ x-x -1)+ _ e ( x+ x)-x x-x is continuous at x=0 , then the… JEE Main 2026 Let α, β ∈ ℝ be such that the function f(x)= cases 2 α(x^ 2 -2)+2 β x &, x be… JEE Main 2026 If f(x)=\ array cc a|x|+x^ 2 -2( |x|)( |x|) x &, x ≠ 0 b &, x=0 array . is continuous at… JEE Main 2026 Let f(x)= cases a x^ 2 +2 a x+3 4 x^ 2 +4 x-3 &, x ≠-(3)/(2), (1)/(2) ~b &, x=-(3)/(2)… JEE Main 2026 Let [·] denote the greatest integer function, and let f(x)= \ 2 x, x^ 2 \ . Let S = \ x… JEE Main 2026 Full Continuity and Differentiability list Continuity and Differentiability JEE Maths PYQs JEE Mathematics PYQs All JEE Main PYQs