Question

If f(x) satisfy f'(x) > 0 , f''(x) > 0 when x (-infinity, 0), f'(x) <0 , f'' (x) > 0 when x (0, infinity)

what graph can it be?

Answer 1

I know that the graph doesn't cross the y axis.
and since f'(x) < 0 , f''(x) > 0 , can i conclude that f(x) < 0?

Answer 2

since the sign of f(x) times the sign of f'(x) gives the sign of f"(x) , that's what i remember from curve sketching

Answer 3

increasing concave up
decreasing concave up

Answer 4

you have no idea where the function is because you don't know any value of \(f\)
it could be positive, negative, zero, whatever

Answer 5

then how can u assume that f is above x axis?

Answer 6

i cannot

Answer 7

oh ok i get it thx

Answer 8

so u are able to determine f''(x) base on f(x) and f'(x), but u cannot determine f(x) base on f'(x) and f''(X)

Answer 9

\(f'\) will tell you the slope of the tangent lines, so combined with \(f''\) which gives concavity you can tell where the function is increasing, decreasing, concave up or concave down. but you cannot tell where the function actually is without knowing values of the actual function
consider a simple case where \(f'(x)=2x\) and \(f''(x)=2\)
this tells me the function is decreasing on \((-\infty,0)\) increasing on \((0,\infty)\)and always concave up
but i have no idea where it is
it could be \(y=x^2\) or \(y=x^2-20\) or \(y=x^2+\pi\) or anything similar.
so while we know what the curve looks like, we don't know where it is

Answer 10

very helpful thanks. i will read it through!