There's something wrong with this question, if the inequality is supposed to hold for all . For example, the function satisfies the hypotheses of the question but not the conclusion.
The best you can realistically ask for is that there exists some neighbourhood of the origin, say , such that whenever .
With that amendment to the question, your proof is partially correct. But there's a defect in it. The problem is that your choice of depends on x. You need to adjust the proof so as to get an estimate of the size of that does not depend on x. This is possible because you are told that f'' is continuous. That means that it is bounded in any closed bounded interval. So given the interval there exists M>0 such that for all x in that interval. You can then use M in place of in your proof, define , and the proof will work nicely.