Let's discuss this objective problem number 14 from ISI BStat BMath. Try to solve the problem and then read their solution.
Problem 14
f(x) = tan(sinx) (x > 0)
To understand the graph of a function, easiest and the most proper method is to apply techniques from calculus. We will quickly compute, derivative and second derivative and try to understand extreme points and convexity of the curve.
$latex f'(x) = cos (x) sec^2 (sin (x)) $
Hence when cos(x) is positive (correspondingly negative), derivative is positive (is negative). Therefore from $ (0, \frac{\pi}{2} ) $ function is increasing, $ ( \frac{\pi}{2} , \frac{3\pi}{2} ) $ the function is decreasing. Also it has critical points on cos (x) is 0 (at $ x = \frac{\pi}{2} , \frac{3\pi}{2} $ )
Now we compute the second derivative.
$ f''(x) = - \sec^2 (\sin(x)) (\sin(x) - 2 \cos^2(x) \tan (\sin(x))) $
At $ x = \frac{\pi}{2} $ second derivative is $ -\sec^2 1 $ [hence we have a maxima] and at $ x = \frac{3\pi}{2} $ second derivative is $ \sec^2 1 $ [hence we have a minima]
Finally we compute $ f(\frac{\pi}{2} ) = \tan 1 > 1 , f(\frac{3\pi}{2} ) = - \tan 1 < -1 $.
Moreover, the function is differentiable at the points of maxima and minima. Hence answer is (B)
Publishing original scientific research in a peer-reviewed journal is an achievement many scholars work toward for years. Achieving this feat early in one's academic journey requires exceptional intellectual rigor, creativity, and persistence. Aaditya Punatar and Muskan Sadana have reached this remarkable milestone, successfully publishing their collaborative research paper in the prestigious Journal of Emerging Investigators […]
Her project on quantum-accelerated Steiner trees for 15-minute cities earned a place among the finalists at the world's largest pre-college STEM competition. Prishaa Shrimali of Marymount School of New York, was selected as a Grand Finalist at the 2026 Regeneron International Science and Engineering Fair (ISEF) in Phoenix, Arizona. Her project was Optimizing Urban Accessibility: […]
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's M.Stat Entrance. They ranked within the first 50 in the entire country in these entrances. I.S.I. M.Stat Entrance
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's B.Stat Entrance and Chennai Mathematical Institute's B.Sc. Math Entrance. They ranked within the first 200 in the entire country in these entrances. Most of these students attended the problem solving workshops regularly, which happen 5 days every week. CMI B.Sc. Math Entrance […]
