3. Let T,(x)= f(a) + f '(a)(x-a)+ "(a 2(x-a)² +² a)(x-a)³ be the Taylor polynomial of degree three for the 2 function f(x), centered at the point x=a. a) Find T,'(x), T, "(x), and T,"(x). b) Evaluate T, '(x), T,"(x), and T,"''(x) at x=a to verify that: T;(a) = f(a), T, '(a) = f '(a), T",(a) = f"(a), T,""(a) = f""(a). and

Let f(x) be a real function and g(x) be a function given by g(x)= f (x)-(f (x))* +(f (x))' for all xeR. Prove that f (x) and g(x) increase or decrease together.

Find all poles, their order and their residues for the function f(z)=(z+1/z-1)3  ( z+1 in numerator and z-1 in denominator whole to power 3)

Consider the function f: R → R defined by f(x) = x³ – 3x + 2. = r3 Then (A) ƒ is increasing on the interval (-0, -1), decreasing on the interval (-1, 1), and increasing on the interval (1, 0). | (B) ƒ is decreasing on the interval (-x, –1), increasing on the interval (-1,1), and decreasing on the interval (1, 0). | (C) f is increasing for all x. (D) ƒ is decreasing for all x. (E) none of the other choices is correct