Question

maxima and minima of function  y=|x+5|+|x-3|

Answer 1

y = | x + 5 |  +  |  x - 3 |

     | x + 5 | =    x + 5        x >=  - 5
                  =  - (x + 5)       x <=  - 5 

    | x + 3 |  =  x + 3     x >= -3
                     - (x+3)   x <= -3

for x <= -5,      y = - (x+5) - (x+3) = - 2 x - 8 
  for  -5 <= x <= -3,    y = ( x + 5 ) - ( x + 3 )  =   2
    for x >= - 3,              y = x + 5 + x + 3 = 2 x + 8 

for x <= -5 , as x increases, y decreases.   lowest value of y = 2  at x = -5
for  -5 <= x <= -3,     y = constant. = 2
for x > -3 ,      as x increases y increases.

there is a minumum value of y between    - 5 <= x <= -3.  
            minimum value of y = 2 

there is no upper bound at any value of x.  y continuously increases or decreases otherwise.