Hello, Some_One!
Most of your interpretations are correct.
Some clarification is needed on two of them. *
Interpret each piece of information, so that you can be able to sketch the graph
of a function $\displaystyle f$ that satisfies the following conditions:
$\displaystyle (a)\;\;f(0) = 0$
$\displaystyle (b)\;\;f''(x) > 0,\;\;x \neq 0$
$\displaystyle (c)\;\;\lim_{x \to 0^} f'(x) \:=\: \infty$
$\displaystyle (c)\;\;\lim_{x \to 0^+} f'(x) \:=\: \infty$
$\displaystyle (d)\;\;\lim_{x \to \infty} f(x) \:=\: \infty$
$\displaystyle (d)\;\;\lim_{x \to \infty} f(x) \:=\: \infty$
My interpretations are as follows:
(a) yintercept at (0,0)
(b) Concave up for all values of $\displaystyle x$, except at $\displaystyle x = 0$ (possibly an asymptote there?)
(c) As you approach 0 from the left, the slope approaches infinity
and so the function gets higher. *
(c) As you approach 0 from the right, the slope approaches negative infinity
and so the function gets lower. *
If we approach from the right, the function gets higher.
(d) As you approach negative infinity (left), the function gets lower and lower.
(d) As you approach positive infinity (right), the function gets higher and higher.
"As we approach 0 from the left, the slope approaches infinity" . . . true!
. . And the curve is rising . . . but not infinitely.
If the slope is becoming infinite, the curve is getting "more vertical".
. . It may not soar off into the sky.
Consider the lower half of a circle. Start at 6 o'clock and move CCW to 3 o'clock. See it?
I suspect that the graph looks like this: Code:

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