Question

If a function f(x) in the domain x E [0,2] is
f(x) = |x-1| + |x²-2x|, then the minimum value is (?) and the maximum value is (??)

Answer 1

consider two cases
\[
0\le x \le 1\\
1<x\leq 2
\]

Answer 2

In the first case
\[
f(x)= 1-x+ 2 x-x^2=-x^2 +x +1
\]

Answer 3

For the second case
\[
f(x) = x-1 +2x-x^2=-x^2+3x-1
\]

Answer 4

Why is that and how can you finish the problem?

Answer 5

@eliesaab I can use the Xv and Yv formulas, but what about the "x" range? Doesn't it interfer? Or I can make sure about the range by calculating the Xv?

Answer 6

@eliesaab the higher Yv value in the two equations will be the maximum, and the other is the minimum?

Answer 7

To me, you need put everything in logic, like this
for |x -1| :

a) = x-1 if x-1 >0, or x >1

b) = 1-x if x-1 <0 , or x <1

Answer 8

for |x^2-2x|

c) = x^2 -2x if x^2 -2x = x(x-2) >0

There are 2 cases here both x > 0 and x -2 >0 ( x>2) makes the product >0

Or both x <0 and x -2 <0 , this makes the product > 0 also. However, we reject this case because it is out of the domain.

d) -(x^2-2x) =2x -x^2 if x(x-2) <0

For this case, we need x >0 and x -2 <0 , that is 0 < x<2 . We need this case.

for x < 0 and x-2 >0 , reject.

Answer 9

Now, combine!!
[0, 1] case b + case d, then we have 1-x - x^2+ 2x = -x^2 +x -1

Answer 10

[1,2] we have case a + case d) that i x-1 +2x -x^2= -x^2+3x-1

Answer 11

For both interval, take derivative to find the max, min

Then, find f(0), f(1) f(2) to find the min.

Answer 12

You may wish to consider the equivalent form, f(x) = |x-1|+x|x-2|. This rather clearly marks where you should look. Isn't it a joy when things are on the boundary and calculus doesn't help?

Answer 13

@loser66 so, both equations gives 5/4 as max. value and f(0),f(1),f(2) gives 1 (it's the minimum,right?) , we have to calculate for 0,1 and 2 because 1 is root for the 1st module and 0 and 2 are roots for the 2nd module?

I didn't quite understand how you got 4 values on the |x²-2x| part... Can you explain me, please?

@tkhunny that's why the minimum values are on (for x) 0,1 and 2?

Answer 14

Look at the graph

Answer 15

Three minimum at x=0,1,and 2. That minimum is 1
at x=.5 and 1.5 we have a maximum of 1.25

Answer 16

@eliesaab I see now.

Thanks guys/girls!