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I read a question and its explanation, and immediately had a doubt. the question was: $y = ax^3 + bx^2 + cx + 5$ is a curve which goes through $A(-2,0)$..blah blah..

d question was to find values of a,b,c

in the explanation, they took out the derivative of equation:

$$y' = 3ax^2 + 2bx + c$$

and in d above equation, they substituted x by -2 & equated it to 0, so it became:

$$0 = 12a -4b + c$$

can this be done? it was as if they treated y' as a function in x separately and assumed that it too passed through the point $(-2,0)$

another way of looking at it is that they figured out that dy/dx at $x= -2$ will be 0. (But I don't think this is correct)

so my question is, if a curve passes through some point, will its derivative also pass through the same point?

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What is the rest of the question (..blah blah)? That might help answer your question. – robjohn Aug 4 '11 at 9:47
I found this question online: The curve $y=ax^3+bx^2+cx+5$ touches the $x$-axis at $P(-2,0)$ and cuts the $y$-axis at the point $Q$ where its gradient is $3$. Find the equation of the curve completely. Would that be the question you read? If so, the word "touches" is significantly different than "goes through". – robjohn Aug 4 '11 at 10:00
yes that is exactly the question. Can you explain how they just equated the derivative of y to 0 after putting x=-2 ? – Rushil Aug 4 '11 at 12:57
oh shit shit!!! Now i got it! touches means that the dy/dx has to be 0!! sorry guys! :-D – Rushil Aug 4 '11 at 13:01
@Rushil: right, but do avoid the profanities next time... :) – J. M. Aug 4 '11 at 13:19

The answer is no. If you're looking for a further explanation of what they were doing or whether they perhaps made a mistake, you'll need to expand on the "blah blah" part. Since the information you've reproduced doesn't fix $a$, $b$ and $c$, presumably that part contains further relevant information?

[Edit in response to the comment:]

This is a good example of why one should quote problems verbatim instead of paraphrasing them. The fact that one has a question about how to solve the problem indicates that there's a chance one doesn't fully understand the problem statement, and paraphrasing things one doesn't fully understand tends to introduce errors.

In the present case, as robjohn had already indicated in a comment, the difference between "touch" and "go through" is crucial. To "touch" means to touch without intersecting. For the curve to merely touch the $x$-axis without intersecting it, the derivative at that point needs to be $0$.

P.S.: I just read the question to the end -- I see now why you may have thought that "touch" means "go through", since it also says the slope is $3$ where it "touches" the $y$-axis, which is impossible in the sense of touching without intersecting. So it seems that (if you now quoted the problem correctly verbatim) they used the word inconsistently -- the fact that they're setting the derivative to $0$ at $-2$ strongly indicates that the first use of "touch" is in the standard sense of "touch without intersecting".

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ok so here is the full question: – Rushil Aug 4 '11 at 12:53
y=ax3+bx2+cx+5 is a curve which touches A(−2,0) and touches the Y-axis at a point where its slope is 3. Find the values of a,b,c Value of c = 3 (that was easy) whatabt the other two? – Rushil Aug 4 '11 at 12:56

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