Relative minimum (maximum) and absolute minimum (maximum)?

Question

Can you please give a simple definintion for relative minimum(maximum) and absolute minimum (maximum). I need to understand the difference between the terms relative and absolute.

Thanks.

Answer 1

Relative min/max means a max or min relative to an interval. For example, if you look at the function x^3-3*x^2+2*x, it has a relative max in the interval (0, 1) (look at the graph). But absolute means it's the max on the entire domain. The above function does not have any absolute maximum because it increases to infinity if you look at the entire domain.

I just remember it like that: absolute max means it is ABSOLUTELY the max (in the sense the word is used in English), while a relative max means relative to the set you're looking at. Kind of like, I'm the shortest person relative to my classmates, but I'm not absolutely the shortest person in the world.

Answer 2

Hello,

Take the equation: (1/4)x^4 +(1/3)x^3 - x^2 = 0

For this equation the y-value increases without bound as x gets large in either the positive or negative direction. Therefore this graph must have a y-value that is an absolute minimum by no y-value can be an absolute maximum. Now if you take the derivative we see that at x=1 there is a minimum because the second derivative is positive and at x = 0 there is a relative maximum because the second derivative is negative and at x = -2 there is a minimum because the second derivative is positive. Now there is no absolute maximum because the graph increases without bound as x -increases in either the positive or negative direction so max. at x=0 is a relative max. Since the graph increases without bound and there are two minimums one or both have to be absolute mins. As it turns out only one is a absolute min . and the other is a relative min. You can check it out, but I believe at x=-2 is the absolute min. and at x=1 is a relative min.

Hope This Helps!!

Answer 3

A relative minimum (or maximum) occurs over a smaller interval of x-values, whereas the absolute maximum or minimum occurs over the entire domain of the function.

For example.

The equation y = x(12 - x)^2 on the domain of [0, 20]

has a relative maximum at approx x = 3.33 .. with a value of approx 148.15 ...but this isn't the highest value of this function over the given domain. The highest value is at the endpoint x = 20 (and is 2000).. so the absolute maximum value is 2000 at x = 20, a relative maximum value is 148.15 at x = 3.33.

A relative maximum occurs when the slope of the graph goes from positive to negative... and this doesn't have to be the highest point on the graph (the example I've given shows this)

Answer 4

Minima and maxima is when the slope is equal to zero. Take for an example the function x^2. The slope at any point is given by the equation 2x (the derivative). At what point does 2x=0 ? Only when x=0. That means the the function x^2 has an absolute minima at x=0. You can find out whether it's a minima or a maxima by seeing whether the slope increases or decreases around the inflection point. For a minima, the slop would have a U shape, increasing on both sides. Sometimes a function has multiple inflection points (where the slope is zero). Some might be a local minima or maxima. That means that while the slope is zero at that point, there are other points where the functions gives a higher or lower value.