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All the critical points are associated with points where either f'(x) = 0 or f''(x) = 0, that is, the roots of the first and second derivative of f, or where one of these doesn't exist.

A local maximum or minimum can occur only where f'(x) is zero or doesn't exist.

f(x) can only increase between a local minimum and a local maximum; it can only decrease between a local maximum and a local minimum; etc.

Concave up and concave down depend on the sign of f''(x). Again, this sign can only change where f''(x) = 0 or where it ceases to exist.

So the first step is to do the symbolic differentiation required to compute f'(x) and f''(x).

Both of these can be represented as fractions. Roots will generally be the points where the numerators are 0; the functions cease to exist at points where the denominators are 0. (Note the special attention needed if there are points where the numerator and denominator are both 0).

With all the roots computed, you can determine the characteristics of each point and each interval.

2007-11-18 00:16:38 · answer #1 · answered by simplicitus 7 · 0 0

You need to find f' & f". Do you know the relations between these points and the derivatives?

2007-11-17 23:08:57 · answer #2 · answered by Tony G 2 · 0 0

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