Examples on Monotonicity Set 5

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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 \(\mathbb{R}\)