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Consider the function $f(x)=\max(7-x,x+3).$ In which range does $f$ take its minimum value$?$

1.  $-6\leq x<-2$
2.  $-2\leq x<2$
3.  $2\leq x<6$
4. $6\leq x<10$
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$y = f(x) = \max\left( 7-x,x+3\right )\rightarrow(1)$

We can draw the graph To find the intersection point we can equate the two lines

$\implies7-x=x+3$
$\implies 2x = 4\implies x = 2$

Put $x = 2$ in equation $(1)$ and get $y = 5$
$\implies$ Intersection point $(x,y)=(2,5)$

Among the given options only option C range includes $2.$

So, Correct answer is $(C).$

edited by

$\rightarrow$ If we try to decrease 7-x to a minimum then x+3 will increase and vice versa since x has opposite sign in both case.

$\rightarrow$ As a result the higher value among 7-x  and x+3  will be selected.

$\rightarrow$ So in order to cancel this effect both values should become same at some point and that should be the required answer.

$\rightarrow$  7-x = x+3 => 2x = 4 => x=2.

$\because$ x=2 comes in the range  2≤x<6

$\therefore$ Option C is the correct answer.