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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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    @Rushil: right, but do avoid the profanities next time... :)2011-08-04

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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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    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?2011-08-04