1
$\begingroup$

Let's $f(x)$ and $g(x)$ be continuous functions on $[0,1]$.

Show an example of the maximum of $f(x)$ and $g(x)$ does not equal to the maximum of $(f+g)(x)$ on $[0,1]$.

Now I have tried to find an example, but for each time, the maximum of $f(x) +$ the maximum of $g(x)$ would equal to the maximum of $(f+g)(x)$. For example, I would have $f(x) = x^2$ and $g(x) = x+2$. The maximum of $f(x)$ would be $1$ and the maximum of $g(x) = 3$, so $1+3 = 4$. However, when I have $(f+g)(x) = x^2 + x + 2$, the maximum of $(f+g)(x) = 4$. This would happen each time I used a continuous function for both $f(x)$ and $g(x)$. Any help would be greatly appreciated.

  • 1
    Hint: You need $f(x)$ and $g(x)$ to have their maxima in different places.2012-10-13
  • 0
    What does "the maximum of f(x) *and* g(x)" mean **exactly** for you?2012-10-13
  • 1
    [Quasi-duplicate](http://math.stackexchange.com/q/209350).2012-10-13
  • 0
    I may be wrong, but when I mentioned maximum here, I am referring to global maximum. I thought the global maximum can be found by the closed interval [0,1] and since 1 is the highest number in the interval, the maximum of f(x) would be 1 and the maximum of g(x) would be 3. Please correct me if I am wrong.2012-10-13
  • 1
    Hint $U+ /\$...2012-10-13
  • 0
    @N.S. Nice hint. Let me suggest $\mathrm V+\Lambda$.2012-10-13

3 Answers 3