Extrema

(Review Exercises) Find the global maximum and minimum (if there are) of the following functions in the given intervals: a) $f(x) = x(x-2)^2$, $[0,+\infty)$ b) $f(x) = x-\arctan x$, $(-\infty,+\infty)$ c) $f(x) = x\lg x$, $[1,2]$ d) $f(x) = \arctan x-\lg(1+x^2)$, $[0,1]$ e) $f(x) = \lg\sin x-2\sin x$, $(0,\pi/2]$ f) $f(x) = 3\lg(1+x^2)+5$, $[-2,0]$
TEST 2 Questions: a) Consider the function $f(x) = xe^{\frac{x-1}{x}}$. Determine whether: i. The function has a global minimum. ii. The function has a local minimum but not a global one. iii. The function has a global maximum. b) Establish if $A = (0,0)$ is a maximum, minimum or a saddle point for $f(x,y) = e^x(2x^2-xy+y^2)$.
Written test 2022-4-18: a) Determine that $x_1 = 2-\sqrt{6}$ is a minimum point and $x_2 = 2+\sqrt{6}$ is a maximum point for $f(x) = \frac{x-2}{x^2+2}$. b) Determine maximum and minimum values of $f(x,y) = x^2+3y$ subject to the constraint $\frac{x^2}{4} + \frac{y^2}{9} = 1$.
Exercises (3-6 April): Find (if there are) the global maximum and the global minimum of the following functions in the given intervals: a) $f(x) = x - \arctan x$, $(-\infty,+\infty)$ b) $f(x) = \sin x - \cos x$, $[0, 2\pi]$ c) $f(x) = \frac{x}{x^2+1}$, $[-2, 3]$