Question

Find the maximum and minimum values, if any of the following function given by
i) $f\left(x\right)=|x+2|-1$
ii) $g\left(x\right)=-|x+1|+3$
iii) $f\left(x\right)=|\sin 4x+3|$

Answer 1

$\left(i\right)f\left(x\right)=|x-2|-1$

Minimum value of $|x-2|=0$

Minimum value of f(x)=minimum value of $|x-2|-1$

$=0-1=-1$

Minimum value of $f\left(x\right)=-1\to (x=-2)$

f(x) has no maximum value.

$\left(ii\right)f\left(x\right)=-|x+1|+3$

$|x+1|>0$

$\Rightarrow -|x+1|<0$

Maximum value of g(x)=maximum value of $-|x+1|+3$

$=0+3=3$

Maximum value of $g\left(x\right)=3$

There is no minimum value of g(x)

$\left(iii\right)f\left(x\right)=|\sin 4x+3|$

$-1\le \sin \theta \le 1$

$-1\le \sin 4x\le 1$

$-1+3\le \sin 4x+3\le 1+3$

$2\le \sin 4x+3\le 4$

$\left|2\right|\le |\sin 4x+3|\le \left|4\right|$

$2\le |\sin 4x+3|\le 4$

$2\le \left|f\right(x\left)\right|\le 4$

Maximum value of $f\left(x\right)=4$

Minimum value of $f\left(x\right)=2$