MathematicsAdvancedHigh School (11–12)Functions and Calculus › Continuity and DifferentiabilityJEE AdvancedOlympiad
Created by StemPy LabA function is differentiable at every real number. All that is known about it is this: , , and its slope always lies between and , that is, for every real . 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?
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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← 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?Each statement below claims an inequality for every x in the open interval 0 < x < π/2. The usual way to prove such a claim is to move everything to one side, call the result u(x), check that u(0) = 0 (or its limit there), and show that u'(x) > 0 on the interval, using the derivative again if needed. Some of the claims below are true and some are false; a single value of x is enough to show that a claim is false. Which of the following statements are true for every x with 0 < x < π/2? →
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