Show that the function given by \(f(x) = 3x + 17\) is increasing on \(\mathbb{R}\).
Show that the function given by \(f(x) = e^{2x}\) is increasing on \(\mathbb{R}\).
Show that the function given by \(f(x) = \sin x\) is
(a) increasing in \((0, \frac{\pi}{2})\)
(b) decreasing in \((\frac{\pi}{2}, \pi)\)
(c) neither increasing nor decreasing in \((0, \pi)\).
Find the intervals in which the function \(f\) given by \(f(x) = 2x^2 - 3x\) is
(a) increasing
(b) decreasing.
(a) \((\frac{3}{4}, \infty)\)
(b) \((-\infty, \frac{3}{4})\)
Find the intervals in which the function \(f\) given by \(f(x) = 2x^3 - 3x^2 - 36x + 7\) is
(a) increasing
(b) decreasing.
(a) \((-\infty, -2)\) and \((3, \infty)\)
(b) \((-2, 3)\)
Find the intervals in which the following functions are strictly increasing or decreasing:
(a) \(x^2 + 2x - 5\)
(b) \(10 - 6x - 2x^2\)
(c) \(-2x^3 - 9x^2 - 12x + 1\)
(d) \(6 - 9x - x^2\)
(e) \((x + 1)^3 (x - 3)^3\)
(a) decreasing for \(x < -1\) and increasing for \(x > -1\)
(b) decreasing for \(x > -\frac{3}{2}\) and increasing for \(x < -\frac{3}{2}\)
(c) increasing for \(-2 < x < -1\) and decreasing for \(x < -2\) and \(x > -1\)
(d) increasing for \(x < -\frac{9}{2}\) and decreasing for \(x > -\frac{9}{2}\)
(e) increasing in \((1, 3)\) and \((3, \infty)\), decreasing in \((-\infty, -1)\) and \((-1, 1)\)
Show that \(y = \log(1 + x) - \frac{2x}{2 + x}\), \(x > -1\), is an increasing function of \(x\) throughout its domain.
Find the values of \(x\) for which \(y = [x(x - 2)]^2\) is an increasing function.
\(0 < x < 1\) and \(x > 2\)
Prove that \(y = \frac{4\sin \theta}{2 + \cos \theta} - \theta\) is an increasing function of \(\theta\) in \([0, \frac{\pi}{2}]\).
Prove that the logarithmic function is increasing on \((0, \infty)\).
Prove that the function \(f\) given by \(f(x) = x^2 - x + 1\) is neither strictly increasing nor decreasing on \((-1, 1)\).
Which of the following functions are decreasing on \((0, \frac{\pi}{2})\)?
(A) \(\cos x\)
(B) \(\cos 2x\)
(C) \(\cos 3x\)
(D) \(\tan x\)
A, B
On which of the following intervals is the function \(f\) given by \(f(x) = x^{100} + \sin x - 1\) decreasing?
(A) \((0, 1)\)
(B) \((\frac{\pi}{2}, \pi)\)
(C) \((0, \frac{\pi}{2})\)
(D) None of these
D
For what values of \(a\) the function \(f\) given by \(f(x) = x^2 + ax + 1\) is increasing on \([1, 2]\)?
\(a > -2\)
Let \(I\) be any interval disjoint from \([-1, 1]\). Prove that the function \(f\) given by \(f(x) = x + \frac{1}{x}\) is increasing on \(I\).
Prove that the function \(f\) given by \(f(x) = \log \sin x\) is increasing on \((0, \frac{\pi}{2})\) and decreasing on \((\frac{\pi}{2}, \pi)\).
Prove that the function given by \(f(x) = \log |\cos x|\) is decreasing on \((0, \frac{\pi}{2})\) and increasing on \((\frac{3\pi}{2}, 2\pi)\).
Prove that the function given by \(f(x) = x^3 - 3x^2 + 3x - 100\) is increasing in \(\mathbb{R}\).
The interval in which \(y = x^2 e^{-x}\) is increasing is
(A) \((-\infty, \infty)\)
(B) \((-2, 0)\)
(C) \((2, \infty)\)
(D) \((0, 2)\)
D