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

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- Jan 24th 2010, 06:49 PMkattdifferences
can someone tell me what is the difference between a relative max. or min. and an absolute max or min value?

- Jan 24th 2010, 07:06 PMVonNemo19
Relative Max: Let $\displaystyle c$ be a critical number of a function $\displaystyle f$.

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

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

Now let $\displaystyle c\in[a,b]$

3. Suppose that $\displaystyle f$ is defined over the interval $\displaystyle [a,b]$ and $\displaystyle a\leq{c}\leq{b}$. Then, if $\displaystyle f(c)\geq{f}(x)$ for all $\displaystyle x\in[a,b]$, $\displaystyle f(c)$ is an absolute maximum for $\displaystyle f$ on $\displaystyle [a,b]$ - Jan 25th 2010, 03:56 AMHallsofIvy
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