Exams › JEE Advanced › Maths

Let f: R -> R and g: R -> R satisfy f(x + y) = f(x) + f(y) + f(x)*f(y) and f(x) = x*g(x) for all x, y in R. Given that lim_(x->0) g(x) = 1, which of the following statements are TRUE?

1. f is differentiable at every x in R
2. If g(0) = 1, then g is differentiable at every x in R
3. The derivative f'(1) equals 1
4. The derivative f'(0) equals 1

Correct answer: f is differentiable at every x in R

Solution

From the functional equation, f(0)=0 and f'(0) = lim f(h)/h = lim g(h) = 1. Since f'(x) = 1*(1+f(x)), f is differentiable everywhere. Statements A and D are TRUE; g is also differentiable everywhere (B TRUE); f'(1) = 1+f(1) which is not generally 1 (C FALSE).

Related JEE Advanced Maths questions

• If f(x) = { (sin⁻¹x)² cos(1/x), x ≠ 0; 0, x = 0 }, then
• For the function f(x) = sin(x) + cos(x) defined on the interval [0, 2π], which of the following is true about its behavior?
• Consider a function f defined on the interval [0, 2] such that it is continuous on [0, 2] and differentiable on (0, 2), with f(0) = 1. Define F(x) = ∫₀ˣ f(√t) dt for x ∈ [0, 2]. If it is given that F'(x) = f'(x) for every x in (0, 2), what is the value of F(2)?
• Consider two continuous functions f and g defined on the interval [-1, 2], which are also twice differentiable on (-1, 2). The values of f and g at x = -1, 0, and 2 are provided in the table below: x = -1 x = 0 x = 2 f(x) = 3 f(x) = 6 f(x) = 0 g(x) = 0 g(x) = 1 g(x) = -1 In the intervals (-1, 0) and (0, 2), the second derivative of (f - 3g), denoted as (f - 3g)'', does not become zero at any point. Which of the following statements is true?
• Suppose f: R → (0, 1) is a continuous function. Which of the following functions equals zero at least at one point within the interval (0, 1)?
• Consider the function f(x) = x + ln(x) − x ln(x), where x lies in the interval (0, ∞). - Column 1 contains details about the zeros of f(x), f'(x), and f''(x). - Column 2 contains information about the behavior of f(x), f'(x), and f''(x) as x approaches infinity. - Column 3 contains details about the increasing or decreasing nature of f(x) and f'(x). Column 1: (I) f(x) = 0 for some x in (1, e²) (II) f'(x) = 0 for some x in (1, e) (III) f'(x) = 0 for some x in (0, 1) (IV) f''(x) = 0 for some x in (1, e) Column 2: (i) lim x→∞ f(x) = 0 (ii) lim x→∞ f(x) = −∞ (iii) lim x→∞ f'(x) = −∞ (iv) lim x→∞ f''(x) = 0 Column 3: (P) f is increasing on (0, 1) (Q) f is decreasing on (e, e²) (R) f' is increasing on (0, 1) (S) f' is decreasing on (e, e²) Which of the following options gives the correct combination?

⚔️ Practice JEE Advanced Maths free + battle 1v1 →