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Question
I have to do a math assignment in preparation for AP Calculus next year and one of the questions is "If f'(x)=3xsin(πx), how many critical points does f(x) have on [0,6]? I have searched the Internet a bit and I think I might have to find the integral? I took precalc last year, but I did not learn how to do this type of problem. If you could offer any sort of assistance it would be greatly appreciated! Thank you!

Answer
Hello again Courtney,

The critical pints of a function are where the first derivative is either zero or undefined.

Thus, there is no need to antidifferentiate f'(x).   Just solve f'(x)=0 on the given interval.
3x*sin(nx)=0 ==> x=0 or sin(nx)=0, the function sin(nx) has a period of 2*pi/n, in other words,
there are n cycles/periods in the space of 2*pi.

If n=1, then sin(1x) is zero at two locations between 0 and 6 (since 6<2*pi)...at x=0 and
x=pi...but we already have x=0...so, for n=1 there are 2 critical points.

If n=2, then sin(2x) there are two cycles in the space of 2*pi, again excluding x=2*pi, since
x is on [0,6], gives 4 critical points.

and so on...there are 2n critical points for f(x) on [0,6] with f'(x)=3x*sin(nx)
-- assuming n is an integer.

Good?  :-)

Abe

Calculus

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Abe Mantell

Expertise

Hello, I am a college professor of mathematics and regularly teach all levels from elementary mathematics through differential equations, and would be happy to assist anyone with such questions!

Experience

Over 15 years teaching at the college level.

Organizations
NCTM, NYSMATYC, AMATYC, MAA, NYSUT, AFT.

Education/Credentials
B.S. in Mathematics from Rensselaer Polytechnic Institute
M.S. (and A.B.D.) in Applied Mathematics from SUNY @ Stony Brook

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