Question

If $x\in [-5,3],$ then the correct option(s) is (are)

• A. $x^3\in [-125,27]$
• B. $x^4\in [0,625]$
• C. $\frac{1}{x}\in [-\frac{1}{5},\frac{1}{3}]$
• D. $\frac{1}{x^2}\in [\frac{1}{25},\infty )$

Answer 1

Answer: (A), (B), (D)

$\frac{1}{x^2}\in [\frac{1}{25},\infty )$

Here, $x\in [-5,3]$
Splitting into intervals
$x\in [-5,0)\cup [0,3]$
$x^3\in [-125,0)\cup [0,27]$
$\Rightarrow x^3\in [-125,27]$

$x^4\in (0,625]\cup [0,81]$
$\Rightarrow x^4\in [0,625]$

$\frac{1}{x}\in (-\infty ,-\frac{1}{5}]\cup [\frac{1}{3},\infty )$

$x^2\in [0,25]$
$\Rightarrow \frac{1}{x^2}\in [\frac{1}{25},\infty )$