1. ## differences

can someone tell me what is the difference between a relative max. or min. and an absolute max or min value?

2. Originally Posted by katt
can someone tell me what is the difference between a relative max. or min. and an absolute max or min value?
Relative Max: Let $c$ be a critical number of a function $f$.

1. If $f'(c)$ from negative to postive at $c$, then $f$ has a relative minimum at $(c,f(c))$

2.If $f'(c)$ from positive to negative at $c$, then $f$ has a relative maximum at $(c,f(c))$

Now let $c\in[a,b]$
3. Suppose that $f$ is defined over the interval $[a,b]$ and $a\leq{c}\leq{b}$. Then, if $f(c)\geq{f}(x)$ for all $x\in[a,b]$, $f(c)$ is an absolute maximum for $f$ on $[a,b]$

3. Originally Posted by katt
can someone tell me what is the difference between a relative max. or min. and an absolute max or min value?
A function, f, has an absolute minimum at x= a if f(a) is less than or equal to any value of f(x).

A function, f, has a relative minimum at x= a if f(a) is less than or equal to any value of f(x) on some small interval around a.

For absolute and relative maximum switch "less than" to "larger than".

What VonNemo19 said about relative maximum and minimum is true for differentiable functions.

But the function f(x)= x for all x except 1, f(1)= 2, has a relative maximum at x= 1 even though the derivative, where ever it is defined, never changes sign