Definition: Let f(k)(0) exist for k = 0,1,2,...,n. Then the nth Taylor polynomial of f about 0 is defined as the polynomial of degree at most n given by
Pa(z) = f(0)+f(0)x+" (0) 2²+(3) (0) + 21 31 + ()(0) n!
a. Show that po has the same value at 0 as ƒ does.
b. Show that p₁ has the same value at 0 and he same slope at 0 as ƒ does.
c. Show that p₂ has the same value at 0, the same slope at 0, and the same concavity at (0,0) as ƒ does.