# Possible functions satisfying this condition

I was solving this question in my assignment:

Question

I didn't have any idea to approach this , so i assumed that the first two are correct and came to the conclusion that the graph would be one of these:

Graph of function

From this , i got [C] and [D] correct too . But i cannot seem to understand why [A] and [B] should be correct always.

let f(x) be a increasing function

i.e. $$f'(x)>0$$ now we have 2 cases to satisfy the given condition

CASE 1 $$f(x)>0$$ and $$f''(x)<0$$ now assume this holds true thus $$f(c)=k$$ where k is positive number

now consider $$f(c-w)$$ such that $$c-w and as $$f'(x)>0$$ therefore $$f(c-w) or $$f(c-w) or $$f(c-w)=k-p$$

we can say there is a '$$w$$'[actually infinately many] such that $$p>k$$ due to which $$f(c-w)<0$$ [which condraticts the assumption of f(x)>0 for all x] as the derivative is always positive and the function keeps decreasing towards the negative side and in fact increasingly decreasing towards negative side

CASE 2 $$f(x)<0$$ and $$f''(x)>0$$ now assume this holds true thus $$f(c)=k$$ where k is negative number

now consider $$f(c+w)$$ such that $$c+w>c$$ and as $$f'(x)>0$$ therefore $$f(c+w)>f(w)$$ or $$f(c+w)>k$$ or $$f(c+w)=k+p$$

we can say there is a '$$w$$'[actually infinately many] such that $$p>|k|$$ due to which $$f(c+w)>0$$ [which condraticts the assumption of f(x)<0 for all x] as the derivative is always positive and the function keeps increasing and in fact increasingly increasing [`increases at a higher rate as value of x increases] as the double derivative is always assumed positive