please tell me how do I find global /local max min using only the first derivative?

Question

how do I know whether it's a global or a local point? I'm so confused

Answer 1

Every hill and mountain on Earth is a local max (higher than surrounding points). The only global max is Everest (the highest point). That's all there is to it.

How to find? Don't learn steps, learn facts. Fermat's Theorem: Max/mins occur at critical points or endpoints. That's all you have to know to find them. To know if they are a max vs a min, use the bold theorem in Bradford T's answer. (Note if that doesn't work, then you can't answer the question with just the first derivative.)

(There is often confusion about stuff like whether a flat top, or also endpoints, can be a max. That depends on how the definition is worded in your book. So don't ask anyone but your teacher. Such questions won't matter after this class anyway because all of it is phrased differently.)

Answer 2

You know how to find critical points of a function using the first derivative then setting the derivative to zero and solving for x. This can give you one or more values. Also, if you have a range given for f(x), [a,b], include f(a) and f(b) in the list of values.

Sort the list of values from the lowest to the highest. The lowest value is the global minimum and the highest value is the global maximum. All the other points are local minimum or maximum points. You will need to perform the first derivative test to determine if the critical are minimum or maximum extrema.

You can have on one global maximum and one global minimum.

Quoting:

If the derivative of a function changes sign around a critical point, the function is said to have a local (relative) extremum at that point. If the derivative changes from positive (increasing function) to negative (decreasing function), the function has a local (relative) maximum at the critical point.

Sort the list of values from the lowest to the highest for each maximum point and each minimum point. The lowest value for the minimums is the global minimum and the highest value for the maximums is the global maximum. All the other points are local minimum or maximum points. You will need to perform the first derivative test to determine if the critical are minimum or maximum extrema.

Remember that the [a,b] endpoints of f(x) could be a global extrema.

Hope this helps.

Answer 3

Taking the first derivative and set it equal to zero, then solve. The will give you your critical values. Then use a sign chart to test points in between the critical values to see if they are positive or negative. If the first derivative is positive, it is increasing if it is negative it is decreasing. You use this to determine if it is a max or min.

negative - critical value - positive is a min

positive - critical value - negative is a max

This picture may help: https://xaktly.com/Images/DerivativesIII/MaxMinFirstDerivativeBox.png