Question

Quick question of critical points of a function.

If I calculate f'(x) = 0 and find the value of x, will this point always be the max or min of the function?

Answer 1

well there are 3 types of stationary points where

f'(x) = 0

they are maximum, minimum and horizontal point of inflection.

to test the point you need to substitute it into the 2nd derivative

so if you have

f'(a) = 0

then if if f"(a) > 0 you have a minimum
f"(a) = 0 a horizontal point of inflection
f"(a) < 0 you have a maximum.

hope it helps

Answer 2

can a function have all 3?

Answer 3

ohh and also what if f'' is a constant then how would you test it?

Answer 4

it can you would need to be degree 3 or higher

Answer 5

well then if you have a constant

then it will be > 0 at all points or < 0 for all points

e.g. f(x) = 2x^2 f'(x) = 4x f"(x) = 4

so you have a stationary point at x = 0 and its a minimum...

Answer 6

points of inflection normally occur between a max and a min

and a point is inflection is where f"(x) = 0

so you can have a value of x where f'(x) = 0 so a stationary point and f"(x) = 0

the stationary point is a horizontal point of inflection

Answer 7

But the mode of a function is always a maximum though correct?

Answer 8

mode of a function.... what is that..?

Answer 9

let me take a picture quick...

Answer 10

So what I am doing is finding the mode of the cumulative d.f. and I know that I need to find f'(x) = 0... so I do that... and I find x = 2... which is all good... but the book tells me I need to also test the points 0 and 3 from the interval... but I do not understand why I am doing this if the f'(x) =0 gives me the max or the mode of this function

Answer 11

well just as a roough guess I'd say it looks like



that's a rough guess... as to what happens... the max value seems to be at x = 3 and the min at x = 0