True or False: If f ''(x) has an absolute minimum at x = a, then f '(a) must exist and have the value 0..

Answer 1

True, if f' is continuous and differentiable at a. But it need not be.

Answer 2

False for the value of 0, I think that f'(a) does have to exist though

for example:

f''(x)=x^2+1
min @x=0
f'(x)=(1/3)x^3+x+b
f'(0)=0+0+b not 0 except if b=0

Answer 3

Are you sure you are asking that correctly? I think the question should be be in regards to f(a) (the function) and not f''(x) (the second derivative).

If the question is about f(x) and f(x) is at a minimum at f(a) (absolute or local for that matter), then the slope of f(a) -- i.e. a line drawn tangent to the curve at a will be flat, correct?

Well, the first derivative of a function IS the slope of that tangent line at a given point, so therefore the f'(a) would be zero because that line at any minima or maxima is flat.

Answer 4

no
it just means that the value of f' at a is the steepest possible slope of the graph

Answer 5

You don't really expect any honest answer for this do you lol