calc question?

Question

Once an extrema (a, b) has been determined, can you tell it is a relative minima if:

a) f '(a) = 0
b) f ' (a) is positive
c) f ' (a) is negative
d) f '' (a) is positive
e) f '' (a) is negative

Answer 1

Here's the thing:

If the first derivative (f ') is either positive or negative, then the function is either increasing or decreasing there. Thus, it will not have a relative minimum or maximum there. In order for a point to be a critical point, f ' must be 0, so we can eliminate b and c.

Now, if f ' is 0, however, this does not mean that we have a relative min or max there. For the former to be true, the function's derivative must change from - to + at that point (hence the function changes from decreasing to increasing at that point). The opposite if true for relative max. Thus, choice a will conclude nothing so we can throw it out.

Now thing about the last two choices. If f '' is positive, the function is concave up...and if f '' is negative the function is concave down. So when some graph is concave up, it is in some way "opening," regardless of whether it is increasing or decreasing. The opposite is true if it is concave down. So what do you think? Would a relative minimum occur when the graph opens up or down? Try to visualize a picture.

Answer 2

Both a and d must be true. If a) was not true, any circle of radius "e" about the point (a,b) would necessarily contain a point (a',b') such that b' < b. For example, the ftn. f(x) = x^2 has f''(2) > 0, but f''(2) is not a relative minimum for any open interval containing 2. The only way that you could have a relative minimum when f'(a) 0 is if the interval in question contains a as an endpoint. For instance, on the interval (2, 3) of the aforementioned function, it is obvious that the rel. minimum occurs at 2 even though f'(2) = 4. (This is why you have to find all places where f' = 0 and also check endpoints when determining extrema.)

Answer 3

i have to believe that the time you took to type this question exceeded the time it would have taken you to look up the answer in your textbook where it describes what convexity means.


(it's d)

Answer 4

d)

Answer 5

d.