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# Y=XSIN(PI X)Y'=SIN(PI X)+ PI XCOS(PI X)I don't know who to solve for the critical numbers because I don't know how to factor this.Thanks!

With other problems such as Y= 2X^3 + 3X^2, I can easily factor the deriviate: Y'= 6X^2 + 6X and set it equal to zero.  With the trig derivative, I don't know how to finish the problem.

### 2 Answers by Expert Tutors

Robert J. | Certified High School AP Calculus and Physics TeacherCertified High School AP Calculus and Ph...
4.6 4.6 (13 lesson ratings) (13)
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Ideas: Critical points are at y' = 0

Therefore, solving y' = sin(πx) + πxcos(πx) = 0 gives you the answer.

How do I find all the x values that give me zero?

There are infinite number of answers. But most likely you are given an interval for x. Therefore, you can use a graphing calculator to get finite number of answers inside the interval.

Y1 = sin(px) + pxcos(px)

Finding zero in the interval gives you all the answers.

Roman C. | Masters of Education Graduate with Mathematics ExpertiseMasters of Education Graduate with Mathe...
4.9 4.9 (364 lesson ratings) (364)
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y=x sin(πx)
y'=sin(πx)+ πx cos(πx)

We want y' to be zero.

so sin(πx)+πx cos(πx) = 0

sin(πx)=-πx cos(πx)

tan(πx)=-πx

x = 0 is one answer. The other don't have analytic expressions so must be solved numerically.

So how do I find all the externa if I can't factor.

Sorry, extrema....

Most graphing calculators have a utility which finds roots of functions. It asks for an initial guess and then uses a built in algorithm (presumably Newton's Method). This utility is usually available when you graph functions.

In your case you would input f(x) = sin(px) + px cos(px).

The p's should have been pi's. I tried using the symbol for pi in the comment, but when submitting it, it replaced them with p's for some reason.