Chapter 6

For all the real values of \(x\), the increasing function is \(\quad\) [PET-1996] (a) \(x^{-1}\) (b) \(x^{2}\) (c) \(x^{3}\) (d) \(x^{4}\)

The function \(f(x)=x^{3}-3 x^{2}-24 x+5\) is an increasing function in the interval given below \(\quad\) IMP PET-1998] (a) \((-\infty,-2) \cup(4, \infty)\) (b) \((-2, \infty)\) (c) \((-2,4)\) (d) \((-\infty, 4)\)

If \(f(x)=x^{3}-10 x^{2}+200 x-10\) then [Kurukshetra CEE-1998] (a) \(f(x)\) decreasing in \(]-\infty, 10]\) and increasing in \([10, \infty)\) (b) \(f(x)\), increasing in \(]-\infty,-10]\) and decreasing in \([10, \infty[\) (c) \(f(x)\) is increasing throughout real line (d) \(f(x)\) is decreasing throughout real line.

The function \(f(x)=x+\cos x\) is [DCE-2002] (a) Always increasing (b) Always decreasing (c) Increasing for certain range of \(x\) (d) None of these

If the function \(f(x)=\cos |x|-2 a x+b\) increasing along entire number scale then the range of ' \(a\) ' is: \(\quad\) [EAMCET-1991] (a) \(a \leq b\) (b) \(a=b / 2\) (c) \(a \leq-1 / 2\) (d) \(a \geq-3 / 2\)

The function \(f(x)=\log (1+x)-\frac{2 x}{2+x}\) is increasing on \(\quad\) [EAMCET-2002] (a) \((-1, \infty)\) (b) \((-\infty, 0)\) (c) \((-\infty, \infty)\) (d) None of these

Let \(f(x)=x^{3}+b x^{2}+c x+d, 0

For all \(x \in(0,1)\) IIIT (Screening)-2000] (a) \(e^{x}<1+x\) (b) \(\log _{e}(1+x)x\) (d) \(\log _{e} x>x\)

If function \(2 x^{2}+3 x-m \log x\) is monotonic decreasing in the interval \((0,1)\) then least value of the parameter ' \(m\) ' is (a) 7 (b) \(15 / 2\) (c) \(31 / 4\) (d) 4

Function \(f(x)=\frac{\lambda \sin x+6 \cos x}{2 \sin x+3 \cos x}\) is increas- ing if [MPPET-2001] (a) \(\lambda>1\) (b) \(\lambda<1\) (c) \(\lambda<4\) (d) \(\lambda>4\)

The function \(f(x)=\cos x-2 a x\) is monotonically decreasing for [RPET-1987; MPPET-2002] (a) \(a<\frac{1}{2}\) (b) \(a>\frac{1}{2}\) (c) \(a<2\) (d) \(a>2\)

The length of a longest interval in which the function \(3 \sin x-4 \sin ^{3} x\) is increasing is IIIT (Screening-2002] (a) \(\pi / 3\) (b) \(\pi / 2\) (c) \(3 \pi / 2\) (d) \(\pi\)

If \(f(x)=k x^{3}-9 x^{2}+9 x+3\) is monotonically increasing in each interval, then (a) \(k<3\) (b) \(k \leq 3\) (c) \(k>3\) (d) None of these

The function \(x^{x}\) is increasing, when [MPPET-2003] (a) \(x>1 / e\) (b) \(x<1 / e\) (c) \(x<0\) (d) For all real \(x\)

The function \(f(x)=\tan ^{-1}(\sin x+\cos x)\) is an increasing function in [AIEEE-2007] (a) \((\pi / 4, \pi / 2)\) (b) \((-\pi / 2, \pi / 4)\) (c) \((0, \pi / 2)\) (d) \((-\pi / 2, \pi / 2)\)

If \(f(x)=\sin x-\cos x\), the function decreasing in \(0 \leq x \leq 2 \pi\) is \mathrm{\\{} I U P S E A T - 2 0 0 1 ] ~ (a) \(\left[\frac{5 \pi}{6}, \frac{3 \pi}{4}\right]\) (b) \(\left[\frac{\pi}{4}, \frac{\pi}{2}\right]\) (c) \(\left[\frac{3 \pi}{2}, \frac{5 \pi}{2}\right]\) (d) None of these

A value of \(c\) for which the conclusion of mean value theorem holds for the function \(f(x)=\) \(\log _{e} x\) on the interval \([1,3]\) is: [AIEEE-2007] (a) \(2 \log _{3} e\) (b) \(\frac{1}{2} \log _{e} 3\) (c) \(\log _{3} e\) (d) \(\log _{e} 3\)

If \(f(x)=\cos x, 0 \leq x \leq \frac{\pi}{2}\), then the real num- ber ' \(\mathrm{c}\) ' for the mean value theorem is [MPPET-1994] (a) \(\pi / 6\) (b) \(\pi / 4\) (c) \(\sin ^{-1}(2 / \pi)\) (d) \(\cos ^{-1}(2 / \pi)\)

The abscissas of the points of the curve \(y=x^{3}\) in the interval \([-2,2]\) where the slope of tangents can be obtained by mean value theorem for the interval \([-2,2]\) are [MPPET-1993] (a) \(\pm 2 / \sqrt{3}\) (b) \(\pm \sqrt{3}\) (c) \(\pm \sqrt{3} / 2\) (d) 0

If \(f(x)=g(x)(x-a)^{2}\) where \(g(a) \neq 0\) and \(g(x)\) is cotinuous at \(x=a .\) Then [Roorkee (Screening)-1999] (a) \(f\) is increasing in the nbd of \(a\) if \(g(x)<0\). (b) \(f\) is decreasing in the nbd of ' \(a\) ' if \(g(a)>0\) (c) \(f\) is increasing in nbd of ' \(a\) ' if \(g(a)>0\) (d) None of these

Let \(f(x)\) and \(g(x)\) be differentiable for \(0 \leq x \leq 1\), such that \(f(0)=0, g(0)=0, f(1)=6\). Let there exist a real number \(c\) in \((0,1)\) such that \(f(c)=\) \(2 g^{\prime}(c)\), then the value of \(g(1)\) must be [Pb. CET-1991] (a) 1 (b) 3 (c) \(-2\) (d) \(-1\)