extrema

The function is given as
y = 7x^3 - 8x^2 -16x +15

The max is at x = -4/7 & min is at x = 4/3

How do I know that it is local max&min or absolute max &min? (graphing calculator is not allowed)

You know that it's a local minimum and maximum because the x^3 term approaches positive infinity as x goes to infinity and negative infinity as x goes to minus infinity.

Generally if the domain or interval is not given, how do we assume the extrema is local or absolute?

If the domain is not given then you can assume it is the maximal possible domain. In this case it is the entire real axis, but in some cases you have to exclude regions where the function is not defined.

You then determine the range of the function. To this this you need to know the properties of the function. In this case it is simple; you are dealing with a polynomial function. The x^3 term dominates for large positive and negative x.

So, you already know by just looking at the function that the range of the function is from minus infinity to plus infinity (note that (-x)^3 = - x^3 ).

This means that the points where the derivatives are zero can't be absolute maxima or minima.

If you have a polynomial of even degree, e.g. with a x^4 term then, depending on the sign of the x^4 term, the range extends either from some minumum value to plus infinity or to negative infinity, but not from minus to plus infinity.

If the range extends to plus infinity, then you can only have local maxima, but a minimum can be global. You should then list al the minima and examine at which of these the value of the function is the least.

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