Let [x] be the greatest integer less than or equal to…

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Let [x] be the greatest integer less than or equal to x, so that x - [x] is the fractional part of x, which lies in [0, 1). Define g(x) = [x] + √(x - [x]) for every real x. On each interval [n, n + 1), with n an integer, this is g(x) = n + √(x - n). Here n stands for any integer, positive, zero or negative, and the square root is always the one that is not negative. The first term jumps by 1 at every integer, and the square root starts again from 0 there. Decide about each statement below on its own. Which of the following statements are true?

MathematicsAdvancedHigh School (11–12)Functions and Calculus › Continuity and Differentiability JEE Advanced

Let $[x]$ be the greatest integer less than or equal to $x$ , so that $x - [x]$ is the fractional part of $x$ , which lies in $[0, 1)$ . Define $\displaystyle g(x) = [x] + \sqrt{x - [x]} \quad \text{for every real } x.$ On each interval $[n, n + 1)$ , with $n$ an integer, this is $g(x) = n + \sqrt{x - n}$ . Here $n$ stands for any integer, positive, zero or negative, and the square root is always the one that is not negative. The first term jumps by $1$ at every integer, and the square root starts again from $0$ there. Decide about each statement below on its own. Which of the following statements are true?

One or more options are correct. Tick all the correct ones: a wrong option costs marks, and a part-right answer earns part of the marks.

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← In each function of List-I, [x] is the greatest integer less than or equal to x. For each function, decide how it behaves at the single point x = 0: whether it is continuous there, whether it is differentiable there, and, if it is, what f'(0) is and whether f' is continuous at 0. Match each function in List-I with the one entry of List-II that describes it. One entry of List-II is not used. List-I P. f(x) = x² [x] Q. f(x) = x² cos1x for x ≠ 0, and f(0) = 0 R. f(x) = x [x] S. f(x) = x e^(|x|) List-II 1. f is continuous at x = 0 but not differentiable there 2. f is differentiable at x = 0, but f' is not continuous at x = 0 3. f is differentiable at x = 0 with f'(0) = 0, and f' is continuous at x = 0 4. f is not continuous at x = 0 5. f is differentiable at x = 0 with f'(0) = 1 A function f is differentiable at every real number. All that is known about it is this: f(0) = 1, f(4) = 9, and its slope always lies between 1 and 3, that is, 1 ≤ f'(x) ≤ 3 for every real x. Many different functions fit these facts, and the statements below are about all of them together. The mean value theorem, used on suitable intervals and from both ends, is the main tool. Think carefully about whether a bound can actually be reached by a differentiable function. Which of the following statements are true? →

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