Exams › JEE Advanced › Maths

Let f(x) be a differentiable function on [2, 5] such that f(2) = 1/5 and f(5) = 1/2. Then there exists x0 with 2 < x0 < 5 such that f'(x0) equals:

1. 1/15
2. 1/10
3. 1/5
4. 1/2

Correct answer: 1/10

Solution

By the Mean Value Theorem: if f is continuous on [2,5] and differentiable on (2,5), there exists x0 in (2,5) such that f'(x0) = (f(5)-f(2))/(5-2) = (1/2 - 1/5)/3 = (5/10-2/10)/3 = (3/10)/3 = 1/10.

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 →