# Examples on Monotonicity Set 5

Go back to  'Applications of Derivatives'

Example - 19

Let f(x) be a real function and g(x) be a function given by

$g\left( x \right) = f\left( x \right) - {\left( {f\left( x \right)} \right)^2} + {\left( {f\left( x \right)} \right)^3}{\text{for all }}x \in \mathbb{R}.$

Prove that f(x) and g(x) increase or decrease together.

Solution: To prove the stated assertion, we must show that for any x, f'(x) and g'(x) have the same sign.

Differentiating the given functional relation in the question, we get:

\begin{align} {g}'\left( x \right) &={f}'\left( x \right)-2f\left( x \right){f}'\left( x \right)+3{{\left( f\left( x \right) \right)}^{2}}{f}'\left( x \right) \\\\ & ={f}'\left( x \right)\left\{ 1-2f\left( x \right)+3{{\left( f\left( x \right) \right)}^{2}} \right\} \\\\ & ={f}'\left( x \right)\left\{ 1-2y+3{{y}^{2}} \right\}(f(x)\,\text{has been substituted by}{\;y}\;\text{for }\text{convenience})\end{align}

To show that f '(x) and g'(x) have the same sign, we must show that $$(3{{y}^{2}}-2y+1)$$ is always positive, no matter what the value of y (or f (x)) is.

Let $$h\left( y \right)=(3{{y}^{2}}-2y+1)$$

Discriminant of $$h\left( y \right)=412=-8<0$$

\begin{align}&\Rightarrow \qquad \text{The parabola for}\;h (y) \text{will not intersect the horizontal axis.}\\\\&\Rightarrow \qquad h (y) > 0\;\text{for all values of}\;y.\\\\&\Rightarrow \qquad 3{y^2}– 2y + 1 > 0\; \text{for all y values.}\\\\&\Rightarrow \qquad\;f '(x)\;\text{ and}\;g'(x) \; \text{have the same sign}\\\\ &\Rightarrow \qquad f (x)\;\text{and}\;g(x) \;\text{increase or decrease together.}\end{align}

Example – 20

Prove that \begin{align}{\rm{sin}} x < x < {\rm{tan}} x\; \forall x \in \left( {0,\frac{\pi }{2}} \right)\end{align}

Solution: We first prove that $$\text{sin }x<x$$ in the given interval.

\begin{align} & \text{Consider }f(x)=x-\text{sin}\,x\qquad \{f(0)\ will\ be\ 0\} \\ & \Rightarrow \qquad \;{f}'(x)=1-cos\,x \end{align}

In $$(0,{\rm{ }}\pi /2), cos x < 1$$

\begin{align} \Rightarrow \qquad &f '(x) = 1 – cos x > 0 \qquad \forall x \in \left( {0,\pi /2} \right)\\\\\Rightarrow \qquad &f (x) \;\text{is increasing on} \left( {0,\pi /2} \right)\\\\ \Rightarrow \qquad &f (x) > f (0) \qquad \qquad \quad \;\; \forall x \in \left( {0,\pi /2} \right)\\\\\Rightarrow \qquad &x – sin x > 0 \qquad \qquad \quad \;\; \forall x \in \left( {0,\pi /2} \right)\\\\ \Rightarrow \qquad & x > sin x \qquad \qquad \qquad \;\; \forall x \in \left( {0,\pi /2} \right) \qquad \qquad \qquad \dots \rm{(i)} \end{align}

Following the proof above, we now construct another function to prove the second part of the inequality:

\begin{align} &g (x) = {\rm{tan}}x – x \qquad \text{{g (0) will be 0}} \\\\\Rightarrow \qquad &g'(x){\rm{ }} = {\rm{ }}se{c^2}x-{\rm{ }}1\end{align}

\begin{align}&\text{Since}\;se{c^2}x > {\rm{ }}1\; {\rm{for}} \;x \in \left( {0,\pi /2} \right)\\\\ \Rightarrow \qquad &g'(x) > 0 \qquad \forall \;x \in \left( {0,\pi /2} \right)\\\\ \Rightarrow \qquad &g (x)\; \text{is increasing on}\; \left( {0,\pi /2} \right)\\\\ \Rightarrow \qquad &g (x) > g(0) \qquad \forall x \in \left( {0,\pi /2} \right)\\\\ \Rightarrow \qquad &tan x – x > 0 \qquad \forall x \in \left( {0,\pi /2} \right)\\\\ \Rightarrow \qquad &tan x > x\qquad \quad \forall x \in \left( {0,\pi /2} \right) \qquad \qquad \qquad \dots(ii) \end{align}

From (i) and (ii),

\begin{align} {\rm{sin}}\,x < x < {\rm{tan}}x \qquad \forall x \in \left( {0,\pi /2} \right)\end{align}

## TRY YOURSELF – II

Q. 1     Use the concepts of monotonicity to determine whether the following statements are true or false:

(a) \begin{align}2x < 3\sin x - x\cos x\;{\rm{for}}\;{\rm{all}}\;x \in \left( {0,\frac{\pi }{2}} \right)\end{align}

(b) \begin{align}1 - x < {e^{ - x}} < 1 - x + \frac{{{x^2}}}{2}\;{\rm{for}}\;{\rm{all}}\;x \ge 0\end{align}

(c) \begin{align}2\sin x + \tan x \le 3x\;{\rm{for}}\;{\rm{all}}\;x \in \left[ {0,\frac{\pi }{2}} \right)\end{align}

(d) \begin{align}\frac{{\tan x}}{{\tan y}} > \frac{y}{x} \end{align}  for all \begin{align}0 < y < x < \frac{\pi }{2}\end{align}

Q. 2     Find the intervals in which the function

$f\left( x \right) = 2{x^2} - \log \left| x \right|,\;x \ne 0$

is monotonically increasing.

Q. 3     For what values of b   is $$f\left( x \right) = \sin x - bx + c$$ is  monotonically decreasing on $$\mathbb{R}$$

Q. 4     Can $$f\left( x \right) = - 2{x^3} + 21{x^2} - 60x + 41$$  attain a negative value in the interval $$\left( { - \infty ,1} \right)?$$

Q. 5     Find the range of values of a for which

$f\left( x \right) = {x^3} + \left( {a + 2} \right){x^2} + 3ax + 5$

is invertible on $$\mathbb{R}$$