How can you sketch a graph of the function f that satisfies these conditions:
f(0) = 0
f"(x)>0 for x not equal to 0
lim f'(x) = ∞ , lim f'(x) = -∞
x → 0 ^{- .............. } x → 0^{+ }^{lim f(x) = -}∞^{, lim f(x)=}^{∞ x}→-∞ ..... x→∞
How can you sketch a graph of the function f that satisfies these conditions:
f(0) = 0
f"(x)>0 for x not equal to 0
lim f'(x) = ∞ , lim f'(x) = -∞
x → 0 ^{- .............. } x → 0^{+ }^{lim f(x) = -}∞^{, lim f(x)=}^{∞ x}→-∞ ..... x→∞
... this means that as x moves further and further to the left (ie gets more and more negative, if you like) y moves further and further down. So the left hand side of the graph will be in the 3rd quadrant (think (neg big number, neg big number)). Opposite as x moves further and further to the right, y gets bigger and bigger, so the graph will end up in the 1st quadrant (think, (pos big number, pos big number)).
Try to draw the graph of y=f(x) now.
Starts at the bottom left corner(south west) in Quad 3, scoops up along the left hand side of the y-axis.
Dot at (0, 0).
"Parabola" shape on right hand side of y-axis, left side approaches y-axis like an asymptote, right hand side heads up north-east.
(not a very mathematical explanation). Draw it and you'll see it meets all the criteria.
This is not necessarily true. In fact, the following specific function is continuous everywhere. Also the graph that you drew can not be a parabola for x > 0, since we need the limit of the slope of f(x) as x approaches $0^+$ to be $-\infty$.As x approaches 0+, f(x) must approach infinity.