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Exercise 17.1

Prove that $\mathrm{f}(\mathrm{x})=\mathrm{ax}+\mathrm{b}$ , where $\mathrm{a}, \mathrm{b}$ are constants and $\mathrm{a}<0$ is a decreasing function on $\mathrm{R}$ .

CBSE, Class 12, Exercise 17.1, Increasing and Decreasing Functions, Maths, RD Sharma

Solution: It is given that, $f(x)=a x+b, a<0$ Suppose $x_{1}, x_{2} \in R$ and $x_{1}>x_{2}$ $\Rightarrow a x_{1}<a x_{2}$ for some $a>0$ $\Rightarrow a x_{1}+b<a x_{2}+b$ for some...

Prove that $\mathrm{f}(\mathrm{x})=\mathrm{ax}+\mathrm{b}$ , where $\mathrm{a}, \mathrm{b}$ are constants and $\mathrm{a}>0$ is an increasing function on $\mathrm{R}$ .

CBSE, Class 12, Exercise 17.1, Increasing and Decreasing Functions, Maths, RD Sharma

Solution: It is given that, $f(x)=a x+b, a>0$ Suppose $x_{1}, x_{2} \in R$ and $x_{1}>x_{2}$ $\Rightarrow a x_{1}>a x_{2}$ for some $a>0$ $\Rightarrow a x_{1}+b>a x_{2}+b$ for some...

Prove that the function $f(x)=\log _{a} x$ is increasing on $(0, \infty)$ if $a>1$ and decreasing on $(0, \infty)$ , if $0<a<1$ .

CBSE, Class 12, Exercise 17.1, Increasing and Decreasing Functions, Maths, RD Sharma

Solution: Case I When $a>1$ Suppose $\mathrm{x}_{1}, \mathrm{x}_{2} \in(0, \infty)$ We have, $\mathrm{x}_{1}<\mathrm{x}_{2}$ $\begin{array}{l} \Rightarrow \log _{e} x_{1}<\log _{e} x_{2} \\...

Prove that the function $f(x)=\log _{e} x$ is increasing on $(0, \infty)$ .

CBSE, Class 12, Exercise 17.1, Increasing and Decreasing Functions, Maths, RD Sharma

Solution: Suppose $x_{1}, x_{2} \in(0, \infty)$ We have, $x_{1}<x_{2}$ $\begin{array}{l} \Rightarrow \log _{\mathrm{e}} \mathrm{x}_{1}<\log _{\mathrm{e}} \mathrm{x}_{2} \\ \Rightarrow...

Prove that $\mathrm{f}(\mathrm{x})=\mathrm{ax}+\mathrm{b}$ , where $\mathrm{a}, \mathrm{b}$ are constants and $\mathrm{a}<0$ is a decreasing function on $\mathrm{R}$ .

Prove that $\mathrm{f}(\mathrm{x})=\mathrm{ax}+\mathrm{b}$ , where $\mathrm{a}, \mathrm{b}$ are constants and $\mathrm{a}>0$ is an increasing function on $\mathrm{R}$ .

Prove that the function $f(x)=\log _{a} x$ is increasing on $(0, \infty)$ if $a>1$ and decreasing on $(0, \infty)$ , if $0<a<1$ .

Prove that the function $f(x)=\log _{e} x$ is increasing on $(0, \infty)$ .

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Prove that , where are constants and is a decreasing function on .

Prove that , where are constants and is an increasing function on .

Prove that the function is increasing on if and decreasing on , if .

Prove that the function is increasing on .

Prove that $\mathrm{f}(\mathrm{x})=\mathrm{ax}+\mathrm{b}$ , where $\mathrm{a}, \mathrm{b}$ are constants and $\mathrm{a}<0$ is a decreasing function on $\mathrm{R}$ .

Prove that $\mathrm{f}(\mathrm{x})=\mathrm{ax}+\mathrm{b}$ , where $\mathrm{a}, \mathrm{b}$ are constants and $\mathrm{a}>0$ is an increasing function on $\mathrm{R}$ .

Prove that the function $f(x)=\log _{a} x$ is increasing on $(0, \infty)$ if $a>1$ and decreasing on $(0, \infty)$ , if $0<a<1$ .

Prove that the function $f(x)=\log _{e} x$ is increasing on $(0, \infty)$ .