Class: 12 Subject: Math Type: Long

Question 1: Show that the function given by \( f(x) = 3x + 17 \) is strictly increasing on \( \mathbb{R} \)....

Class: 12 Subject: Math Type: Long

Question 2: Show that the function given by \( f(x) = e^{2x} \) is strictly increasing on \( \mathbb{R} \)....

Class: 12 Subject: Math Type: Long

Question 3: Show that the function given by \( f(x) = \sin x \) is strictly increasing on \( (0, \frac{\pi}{2}) \), strictly decreasing on \( (\frac{\pi}{2}, \pi) \), and neither increasing nor decreasing on \( (0, \pi) \)....

Class: 12 Subject: Math Type: Long

Question 4: Find the intervals in which the function \( f(x) = 2x^2 - 3x \) is strictly increasing and strictly decreasing....

Class: 12 Subject: Math Type: Long

Question 5: Find the intervals in which the function \( f(x) = 2x^2 - 3x^2 - 36x + 7 \) is strictly increasing and strictly decreasing....

Class: 12 Subject: Math Type: Long

Question 6: Show that \( y = \log (1 + x) - \frac{2x}{2 + x},\,x > -1 \), is an increasing function throughout its domain....

Class: 12 Subject: Math Type: Long

Question 7: Find the values of \( x \) for which \( y = [x(x - 2)]^2 \) is an increasing function....

Class: 12 Subject: Math Type: Long

Question 8: Prove that \( y = \frac{4\sin \theta}{(2 + \cos \theta)} - \theta \) is an increasing function of \( \theta \) in \( \left[ 0, \frac{\pi}{2} \right] \)....

Class: 12 Subject: Math Type: Long

Question 9: Prove that the logarithmic function is strictly increasing on \( (0, \infty) \)....

Class: 12 Subject: Math Type: Long

Question 10: Prove that the function \( f(x) = x^2 - x + 1 \) is neither strictly increasing nor strictly decreasing on \( (-1, 1) \)....

Class: 12 Subject: Math Type: Objective

Question 11: Which of the following functions are strictly decreasing on \( \left( 0, \frac{\pi}{2} \right) \)? (A) \( \cos x \) (B) \( \cos 2x \) (C) \( \cos 3x \) (D) \( \tan x \)...

Class: 12 Subject: Math Type: Objective

Question 12: On which of the following intervals is the function \( f(x) = x^{100} + \sin x - 1 \) strictly decreasing? (A) \( (0,1) \) (B) \( \left( \frac{\pi}{2}, \pi \right) \) (C) \( \left( 0, \frac{\pi}{2} \right) \) (D) None of these...

Class: 12 Subject: Math Type: Long

Question 13: Find the least value of \( a \) such that the function \( f(x) = x^2 + ax + 1 \) is strictly increasing on \( (1,2) \)....

Class: 12 Subject: Math Type: Long

Question 14: Let \( I \) be any interval disjoint from \( (-1,1) \), prove that the function \( f(x) = x + \frac{1}{x} \) is strictly increasing on \( I \)....

Class: 12 Subject: Math Type: Long

Question 15: Prove that the function \( f(x) = \log \sin x \) is strictly increasing on \( \left( 0, \frac{\pi}{2} \right) \) and strictly decreasing on \( \left( \frac{\pi}{2}, \pi \right) \)....

Class: 12 Subject: Math Type: Long

Question 16: Prove that the function \( f(x) = \log \cos x \) is strictly decreasing on \( \left( 0, \frac{\pi}{2} \right) \) and strictly increasing on \( \left( \frac{\pi}{2}, \pi \right) \)....

Class: 12 Subject: Math Type: Long

Question 17: Prove that the function \( f(x) = x^3 - 3x^2 + 3x - 100 \) is increasing on \( \mathbb{R} \)....

Class: 12 Subject: Math Type: Objective

Question 18: The interval in which \( y = x^2 e^{-x} \) is increasing is (A) \( (-\infty, \infty) \) (B) \( (-2, 0) \) (C) \( (2, \infty) \) (D) \( (0, 2) \)...

Class: 12 Subject: Math Type: Mcq Year: 2024

Question 19: The absolute maximum value of function \( f(x) = x^3 - 3x + 2 \) in \([0, 2]\) is:...