Monotonic increasing and decreasing functions — practice questions

Question 1 · difficulty L2 · understanding

The function $$f(x)=\frac{x}{x^2-6 x-16}, x \in \mathbb{R}-\{-2,8\}$$

• A. decreases in $$(-\infty,-2) \cup(-2,8) \cup(8, \infty)$$
• B. increases in $$(-\infty,-2) \cup(-2,8) \cup(8, \infty)$$
• C. decreases in $$(-2,8)$$ and increases in $$(-\infty,-2) \cup(8, \infty)$$
• D. decreases in $$(-\infty,-2)$$ and increases in $$(8, \infty)$$

Question 2 · difficulty L3 · understanding

Consider the following three statements for the function $f:(0, \infty) \rightarrow \mathbb{R}$ defined by $f(x)=\left|\log _e x\right|-|x-1|$ : (I) $f$ is differentiable at all $x>0$. (II) $f$ is increasing in $(0,1)$. (III) $f$ is decreasing in $(1, \infty)$. Then.

• A. Only (I) is TRUE.
• B. Only (I) and (III) are TRUE.
• C. Only (II) and (III) are TRUE.
• D. All (I), (II) and (III) are TRUE.

Question 3 · difficulty L3 · understanding

Let the function $f(x) = \frac{x}{3} + \frac{3}{x} + 3, x \neq 0$ be strictly increasing in $(-\infty, \alpha_1) \cup (\alpha_2, \infty)$ and strictly decreasing in $(\alpha_3, \alpha_4) \cup (\alpha_4, \alpha_5)$. Then $\sum\limits_{i=1}^{5} \alpha_i^2$ is equal to

• A. 48
• B. 40
• C. 36
• D. 28

Question 4 · difficulty L3 · understanding

Let $(2,3)$ be the largest open interval in which the function $f(x)=2 \log _{\mathrm{e}}(x-2)-x^2+a x+1$ is strictly increasing and (b, c) be the largest open interval, in which the function $\mathrm{g}(x)=(x-1)^3(x+2-\mathrm{a})^2$ is strictly decreasing. Then $100(\mathrm{a}+\mathrm{b}-\mathrm{c})$ is equal to :

• A. 360
• B. 420
• C. 160
• D. 280

Question 5 · difficulty L3 · understanding

For the function $$f(x)=(\cos x)-x+1, x \in \mathbb{R}$$, between the following two statements (S1) $$f(x)=0$$ for only one value of $$x$$ in $$[0, \pi]$$. (S2) $$f(x)$$ is decreasing in $$\left[0, \frac{\pi}{2}\right]$$ and increasing in $$\left[\frac{\pi}{2}, \pi\right]$$.

• A. Both (S1) and (S2) are incorrect.
• B. Only (S1) is correct.
• C. Only (S2) is correct.
• D. Both (S1) and (S2) are correct.

Question 6 · difficulty L3 · understanding

The interval in which the function $$f(x)=x^x, x>0$$, is strictly increasing is

• A. $$(0, \infty)$$
• B. $$\left(0, \frac{1}{e}\right]$$
• C. $$\left[\frac{1}{e^2}, 1\right)$$
• D. $$\left[\frac{1}{e}, \infty\right)$$

Question 7 · difficulty L3 · understanding

For the function $$f(x)=\sin x+3 x-\frac{2}{\pi}\left(x^2+x\right), \text { where } x \in\left[0, \frac{\pi}{2}\right],$$ consider the following two statements : (I) $$f$$ is increasing in $$\left(0, \frac{\pi}{2}\right)$$. (II) $$f^{\prime}$$ is decreasing in $$\left(0, \frac{\pi}{2}\right)$$. Between the above two statements,

• A. only (I) is true.
• B. both (I) and (II) are true.
• C. only (II) is true.
• D. neither (I) nor (II) is true.

Question 8 · difficulty L3 · understanding

If $5 f(x)+4 f\left(\frac{1}{x}\right)=x^2-2, \forall x \neq 0$ and $y=9 x^2 f(x)$, then $y$ is strictly increasing in :

• A. $\left(0, \frac{1}{\sqrt{5}}\right) \cup\left(\frac{1}{\sqrt{5}}, \infty\right)$
• B. $\left(-\frac{1}{\sqrt{5}}, 0\right) \cup\left(\frac{1}{\sqrt{5}}, \infty\right)$
• C. $\left(-\frac{1}{\sqrt{5}}, 0\right) \cup\left(0, \frac{1}{\sqrt{5}}\right)$
• D. $\left(-\infty, \frac{1}{\sqrt{5}}\right) \cup\left(0, \frac{1}{\sqrt{5}}\right)$

Question 9 · difficulty L3 · understanding

Let $$f: \rightarrow \mathbb{R} \rightarrow(0, \infty)$$ be strictly increasing function such that \lim _\limits{x \rightarrow \infty} \frac{f(7 x)}{f(x)}=1. Then, the value of \lim _\limits{x \rightarrow \infty}\left[\frac{f(5 x)}{f(x)}-1\right] is equal to

• A. 0
• B. 4
• C. 1
• D. 7/5

Question 10 · difficulty L3 · understanding

Consider the function $$f:\left[\frac{1}{2}, 1\right] \rightarrow \mathbb{R}$$ defined by $$f(x)=4 \sqrt{2} x^3-3 \sqrt{2} x-1$$. Consider the statements (I) The curve $$y=f(x)$$ intersects the $$x$$-axis exactly at one point. (II) The curve $$y=f(x)$$ intersects the $$x$$-axis at $$x=\cos \frac{\pi}{12}$$. Then

• A. Both (I) and (II) are correct.
• B. Only (I) is correct.
• C. Both (I) and (II) are incorrect.
• D. Only (II) is correct.