Recitation Section Written Homework 6 Write legibly and show all your work. You

Question

Recitation Section Written Homework 6 Write legibly and show all your work. You may work together, use your book/notes, and receive guidance from your recitation instructor.H own solutions. The grader has the right to refuse to grade sloppy work. In this problem, you will follow the procedure outlined in the graphing document (for polynomials and functions involving sine and cosine) to sketch the graph of f(x)-+2 sinr on the interval [0,2]. If you do not follow the procedure, expect a very low grade. 1. Find the critical point(s) (r-coordinate only). Your answer(s) must be exact, and you must show your work. (25 points)

Explanation / Answer

Solution:

f(x)=x+2sinx

so, f'(x)=1+2cosx

Now, f'(x)=0 gives, 1+2cosx=0 2cosx=-1 cosx=-1/2

We know, cos(2/3)= -1/2 and cos=cos(+2k) ; where k= 0, ±1, ±2, ±3,......... (i.e, k is any integer).

Therefore, cosx= -1/2 gives, x = 2/3+2k i.e, x = (k+1/3)2

i.e, x = (3k+1)2/3 ; k= 0, ±1, ±2, ±3,.... ...

Therefore, the critical points are: (3k+1)2/3 : k= 0, ±1, ±2, ±3, ±4,.... ......

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