Problem 5 (10 points). Consider the function f(x) = L. (1-4e+") dt, defined in (-3, +00)....
Problem 3. (10 points) For the function f(x,y) = r? - Ty + y2 – 21+ y, find all the critical point(s) and investigate whether it is (or they are) a saddle, local max or local min.
3. (28 points) Let f(x,y) = 2x3 - 6xy+3y- be a function defined on xy-plane. (a) (6 pnts) Find first and second partial derivatives of f. (b) (10 pnts ) Determine the local extreme points of f (max., min., saddle points) if there is any. (C) (12 pnts) Find the maximum and minimum values of f over the closed region bounded by the lines y = -x, y = 1 and y=r
(1 point) Consider the function f(x) = -22% + 36x? - 162x + 10. This function has two critical numbers A <B: А 3 and B 9 f"(A) 36 f"(B) = -36 Thus f(x) has a local -206 and a local 10 at A (type in MAX or MIN) at B (type in MAX or MIN).
2. Let f(x,y) = 2x2 - 6xy + 3y2 be a function defined on xy-plane (a) Find first and second partial derivatives of (b) Determine the local extreme points off (max., min., saddle points) if there are any. (c) Find the absolute max. and absolute min. values of f over the closed region bounded by the lines x= 1, y = 0, and y = x
(1 point) For the function f(x) = e2x + e- defined on the interval (-4, o), find all intervals where the function is strictly increasing or strictly decreasing. Your intervals should be as large as possible. f is strictly increasing on f is strictly decreasing on (Give your answer as an interval or a list of intervals, e.g., (-infinity,8] or (1,5),(7,10)) whenever r is near c on the left Find and classify all local max's and min's. (For the purposes...
consider the function f(x,y)=x^2-2xy+3y^2-8y (a)find the critical points of f and classify each critical point as local max min or saddle point (b) does f have a global max ?if so what is it ? does f have a global min ? if so what is it ?
6. Consider the function f(x) = x3 - 10x (a) (3 pts) Find f '(x) (b) (9 pts.) Find the intervals where f(x) is increasing/decreasing, and classify any local max/min. (c) (3 pts) Find f '(x) (d) (9 pts.) Find the intervals where f(x) is concave up/down and classify any inflection points. Using the information from parts a-d only, sketch the graph of y=f(x).
9. [7 points) Consider the function f(x) defined by f(x) = xeAs + B if x <3 C(x - 3)2 if 3 < x < 5 130 if > 5. C Suppose f(x) satisfies all of the following: f(x) is continuous at x = 3. • lim f(x) = 2 + lim f(x). 3+5+ 3-5- lim f(x) = -4. Find the values of A, B, and C. . 24-O
Consider the function f defined on the interval-5, 5 as follows, { E5,0), те (0,5). 3. f(x) = 3. Denote by fr the Fourier series expansion of fon [-5,5]. fF(x)= 2 +b sin а, cos Find the coefficients a, an and b with n> 1. 0 an b = (10-10(-1)^n)/(pin) M M M
Consider the function f defined on the interval-5, 5 as follows, { E5,0), те (0,5). 3. f(x) = 3. Denote by fr the Fourier series expansion of...
Problem 1. Consider the function f(x)- 3.12 show that f is Riemann integrable on [0.2] and use the definition to find .后f(x)dr Problem 2. Consider the function -2, zEQ 2, O f(r) = Show that f is not Riemann integrable on 0,1 but s Reemann integrable on this interval. Problem 3. (a) Let f be a real-valued function on a, b] such thatf()0 for all c, where c E [a, b Prove that f is Riemann integrable on a, b...