Question

2. for the function f(x)= x+2 cos x on the interval [0,2pi] a. find the first derivative
b.) find the second derivative
c.) find the functions critical values(if any). include their y- coordinates in your answers in order to form critical points.
d. )find the intervals on which f is increasing or decreasing.
e. )find the local extrema of f.
f. )find the functions hyper critical values(if any). include their y coordinates
g.) find the intervals of concavity, i.e. the intervals on which the functions is concave up or concave down.
h.) list all the points (both coordinates) that are in flection points.

2. For the function f(x) = x + 2 cos x on the interval [0,21] a) Find the first derivative. b) Find the second derivative. c) Find the function's critical value(s), (if any.) Include their y-coordinates in your answers in order to form critical points. (For y-coordinates, decimals to one decimal place are acceptable.) d) Find the intervals on which f is increasing or decreasing. e) Find the local extrema of f. f) Find the function's hyper-critical value(s), (if any.) Include their y-coordinates. (For y- coordinates, decimals to one decimal place are acceptable.) g) Find the intervals of concavity. i.e. the intervals on which the function is concave up or concave down. h) List all points (both coordinates) that are inflection points.

Answer 1

please upvote. If any doubt, feel free to comment. I will resolve your query asap.

f(x) = x + 2 cosx. on [0, 217]. 2 ces 20) = = 1-2 sinx da (a) f'(x) = f(-) = (x + 2.04 * (6) f(a) f(a) og (1 - 2 un 2) -2002. (c) To find critical values, f'(x) = 0. 1-2 sinx =o. & sinx = 1/2 * * = 5 at x , y=f() = +2 Cos - 262.3 at x Say = f(x) = sq + 2 CS SET = 0.9 cutical points are (1, 2.0) and (5,0.9) citical points on (7.2.3) and (,0.9) cuit

20 = (d){(@) f"(@) at x = f(3) = -13.(-ve) Ko is a point of mascimal f“ (20) at x = sa f" (57) - vB (tve) x asan is a point of minima = function is decreasing between [ S] and increasing between [o. II] and [ SIT , 27] een

(f) hyper-critical point, f'(x) = 0 coroo. -2 cos x = 0 ar oo. : cos x to cosx = 0. - X 20 = II. 34 .3J 2 at x y +2001 (?) = 1.6 at x = PTY = 3.4 +200 (31) = 4.7 - Hypercritical points are (37,1.6) and ( 37, 4.7) | 1 (9)& (h) Since there is a maxima at x = I and minima at x = ST

Since, f"(x) = 0 at x = 37, 37. x = I and 31 are inflection points - Point of inflection on (I.1.9) and (22.4.1) I lies between I and St. The function is concave down in region (0. and concave up in region [177 307] since point of inflection divides concave up and concave down region nce