If a, b be two fixed positive integers such that f(a + x) = b + {[{b^3} + 1

English

Gujarati

Hindi

1.Relation and Function

hard

If $a, b$ be two fixed positive integers such that $f(a + x) = b + {[{b^3} + 1 - 3{b^2}f(x) + 3b{\{ f(x)\} ^2} - {\{ f(x)\} ^3}]^{\frac{1}{3}}}$ for all real $x$, then $f(x)$ is a periodic function with period

$a$

$2a$

$b$

$2b$

Solution

(b) $f(a + x) = b + {(1 + {\{ b – f(x)\} ^3})^{1/3}}$
==> $f(a + x) – b = {\{ 1 – {\{ f(x) – b\} ^3}\} ^{1/3}}$
==> $\phi (a + x) = {\{ 1 – {\{ \phi (x)\} ^3}\} ^{1/3}}$, [$\phi (x) = f(x) – b$]
==> $\phi (x + 2a) = {\{ 1 – {\{ \phi (x + a)\} ^3}\} ^{1/3}} = \phi (x)$
==> $f(x + 2a) – b = f(x) – b \Rightarrow f(x + 2a) = f(x)$
$\therefore$ $f(x)$ is periodic with period $2a$.

Standard 12

Mathematics

Which of the following is function

normal

The domain of the function $f(x)=\frac{1}{\sqrt{[x]^2-3[x]-10}}$ is (where $[x]$ denotes the greatest integer less than or equal to $x$ )

hard

(JEE MAIN-2023)

Set of all values of $x$ satisfying

$\frac{{{x^4} – 4{x^3} + 3{x^2}}}{{({x^2} – 4)({x^2} – 7x + 10)}} \ge 0$

normal

Let $f(x) = cos(\sqrt P \,x),$ where $P = [\lambda], ([.]$ is $G.I.F.)$ If the period of $f(x)$ is $\pi$. then

normal

The function $f\left( x \right) = \left| {\sin \,4x} \right| + \left| {\cos \,2x} \right|$, is a periodic function with period

hard

(JEE MAIN-2014)

If $a, b$ be two fixed positive integers such that $f(a + x) = b + {[{b^3} + 1 - 3{b^2}f(x) + 3b{\{ f(x)\} ^2} - {\{ f(x)\} ^3}]^{\frac{1}{3}}}$ for all real $x$, then $f(x)$ is a periodic function with period

Solution

Similar Questions

Which of the following is function

The domain of the function $f(x)=\frac{1}{\sqrt{[x]^2-3[x]-10}}$ is (where $[x]$ denotes the greatest integer less than or equal to $x$ )

Set of all values of $x$ satisfying $\frac{{{x^4} – 4{x^3} + 3{x^2}}}{{({x^2} – 4)({x^2} – 7x + 10)}} \ge 0$

Let $f(x) = cos(\sqrt P \,x),$ where $P = [\lambda], ([.]$ is $G.I.F.)$ If the period of $f(x)$ is $\pi$. then

The function $f\left( x \right) = \left| {\sin \,4x} \right| + \left| {\cos \,2x} \right|$, is a periodic function with period

Start a Free Trial Now

Set of all values of $x$ satisfying

$\frac{{{x^4} – 4{x^3} + 3{x^2}}}{{({x^2} – 4)({x^2} – 7x + 10)}} \ge 0$