Taylor series for the function f(x)=5cosx, at x= is: Taylor Series (k) (a) 8 Σ k=0 ..... + OA OB. Oc - (x − a) * =ƒ (a) +ƒ'(a) (x − a) + OD. f(n) (a) n! )=5+ f(x) = -5+ = (x-x)² 5 -5- f(x)=5- (x-a)" +.... f"l(a) 2! (x-a) ² + flll (a) 3! (x-a) ³+

1 g(x) = (7 +x)3 g(x) = 2 5 „n (-1)"(n+1)(n + 2) 2.(7)"+3 n=0 Part 3 Use your answers above to now express the function as a power series (centered at x = 0). x2 h(x) = (7 + x)3 h(x) = n=0

The function f is defined by the power series (-1)" x²n f(x) = Σ n=0 (2n+1)! for all real numbers x. Use the first and second derivative test by finding f'(x) and f"(x). Determine whether f has a local maximum, a local minimum, or neither at x=0. x4 1+1-1 5! = 1 −²+ 3! 7! +. + (-1)"x²n (2n+1)!