The maximum and the minimum values, if any of the following functions f (x) = –| x – 1 | + 5 for all

We have

f (x) = –|x – 1| + 5 for all

Now,

 for all

  for all

  for all

f (x)  5 for all

So, 5 is the maximum value of f (x)

Now,

f (x) = 5  – |x –1|+ 5 = 5 |x – 1|

= 0  x = 1

Thus f (x) attains the maximum value 5 at

x = 1

Since f (x) can be made as small as we please therefore the minimum value of f (x) does not exist