Question by ituneschallenged: What is the difference between absolute and local mins and maxs in calculus?
In class, the section is on absolute and local minimas and maximas as related to curve sketching and Rolle’s and the mean value theorem. How can you tell whether it is a local or an absolute?
Best answer:
Answer by Benjamin
anything that is either a peak or a low point of a curve is a local min or max no matter how close another one is (any time the slope changes sign) and only the highest and lowest points of the whole function are the absolute
Give your answer to this question below!
If y = f(x), a local min/max point is the min/max value of f(x) within a certain “local” range of x, whereas an absolute min/max point is the min/max value f(x) over all values of x .
The beauty is, a local max or min can be the absolute max or min. But some times absolute max or min need not be a local max or min. Why so?
Local max or min is defined as the value max or min compared to the neighbourhood points on both sides. So it is certain that the value of the function will be less definitely by the neighbour hood in case of local max and will be more definitely by the neighbourhood in case of local min. Hence the slope of the tangent will become 0 ie the tangent becomes parallel to X-axis. So its first derivative is equated to zero to find the critical number. Hence the stationary point. But if the absolute max or min is found as the max or min value at the extreme values of intervals(closed), then it need not be the local max or min as the first derivative is not found to be 0 at these extreme values.
Now to find out whether it is local max or min, we find the second derivative of the function and find its value at critical number. If it is negative then it is local max and if it is positive then it will be local minimum.
All these will be stated using mathematical symbols.
For example, instead saying x is negative, mathematician states that fact as x< 0.