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Opitmization question
- Thread starter think_bot
- Start date
If f(x) is a differentiable function over (a, b) and neither f(a) nor f(b) is a local minimum, there exists c such that a < c < b, f(c) is a local minimum, and f'(c) = 0.How are Finding the max and min of a optimization problem different. and how do you distinguish them in an equation?
If f(x) is a differentiable function over (a, b) and neither f(a) nor f(b) is a local maximum, there exists d such that, a < d < b, f(d) is a local maximum, and f'(d) = 0.
Notice that f'(e) = 0 and a < e < b is not sufficient to determine whether f(e) is a local maximum or minimum. There are helpful tests. If
f'(e) = 0 and f''(e) > 0, then f(e) is not a maximum. If f'(e) = 0 and f''(e) < 0, then f(e) is not a minimum. But the ultimate test is to evaluate f(a), f(b), and every value in between where the first derivative of f(x) is zero to find the local minimum and local maximum.
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