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A relative (or local) maximum, for a function, is a point such that the value of the function at that point is greater than the values within a region on either side of it. It need not be a global maximum.
For example, consider a functions such as f(x) = 3*x - x^3
[it is shaped a bit like the letter S on its side].
Now f(1) = 2 is greater than all values of f(x) for x > -2. So the point (1, f(1)0 represents a relative maximum. However, for any x less than -2, f(x) is greater than f(1), and as x becomes more and more negative, f(x) becomes infinitely large. So f(1) cannot be a global maximum.
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Does a differentiable function have to have a relative minimum between any two relative maximum?
Yes.
A fisherman travels downstream at full speed to?
The maximum speed that a vessel will achieve relative to ground is its own maximum speed through water plus the speed of the the moving water downstream.
If f is continuous for all x and y and has two relative minima then f must have at least one relative maximum?
Yes.If you find 2 relative minima and the function is continuous, there must be exactly one point between these minima with the highest value in that interval. This point is a relative maxima.Think of temperature for example (it is continuous).
Does every graph of a cubic function have relative maximum and minimum points?
No, since the equation could be y = x3 (or something similar) which will have a point of inflection at (0,0), meaning there is no relative maximum/minimum, as the graph doesn't double back on itself For those that are unfamiliar with a point of inflection <http://mathsfirst.massey.ac.nz/Calculus/SignsOfDer/images/Introduction/POIinc.png>
Use the first derivative test to determine the relative maximum and relative minimum values of y equals 2x3 plus 3x2-36x?
first derivative = 6x2 + 6x - 36 factors are (6x + 18)(x - 2)
