MHB Relative Extrema vs Absolute Extrema

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Relative extrema refer to the highest or lowest points within a specific neighborhood of a function, while absolute extrema represent the highest or lowest points across the entire domain. Finding absolute extrema is more complex because it requires evaluating critical points, poles, and boundary behavior of the function. For example, a function may have multiple relative minima and maxima, but only one absolute minimum, especially if it approaches infinity at the boundaries. Understanding these differences is crucial for analyzing the behavior of functions in calculus. Overall, the distinction between relative and absolute extrema is fundamental in optimization problems.
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In basic terms, what is the main difference between relative extrema and absolute extrema? I know that absolute extrema is more involved but why is this the case?
 
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A relative maximum (minimum) is the greatest (smallest) in its neighborhood but an absolute maximum (minimum) is the greatest (smallest) anywhere (in the domain). For example:

View attachment 6470

In the picture we see that the function has $3$ relative minima and an absolute minimum, which is smallest of all relative minima. It also has $2$ relative maxima, but it has no absolute maximum, since at the boundaries the function goes to $+\infty$.
In general, to find an absolute extrema, besides the critical points we have to check also the poles of the function and the boundaries of its domain.
 

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mathmari said:
A relative maximum (minimum) is the greatest (smallest) in its neighborhood but an absolute maximum (minimum) is the greatest (smallest) anywhere (in the domain). For example:



In the picture we see that the function has $3$ relative minima and an absolute minimum, which is smallest of all relative minima. It also has $2$ relative maxima, but it has no absolute maximum, since at the boundaries the function goes to $+\infty$.
In general, to find an absolute extrema, besides the critical points we have to check also the poles of the function and the boundaries of its domain.

Very cool. Nice picture.
 

Thread 'Proving that convexity implies second order derivative being positive'

There are probably loads of proofs of this online, but I do not want to cheat. Here is my attempt: Convexity says that $$f(\lambda a + (1-\lambda)b) \leq \lambda f(a) + (1-\lambda) f(b)$$ $$f(b + \lambda(a-b)) \leq f(b) + \lambda (f(a) - f(b))$$ We know from the intermediate value theorem that there exists a ##c \in (b,a)## such that $$\frac{f(a) - f(b)}{a-b} = f'(c).$$ Hence $$f(b + \lambda(a-b)) \leq f(b) + \lambda (a - b) f'(c))$$ $$\frac{f(b + \lambda(a-b)) - f(b)}{\lambda(a-b)}...
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