The linearizationL(x) is the best linear(first-degree) approximation to f(x) near x=a, because L(x) and f(x) have the same value and the same rate of change (derivative) at x=a. But we can get a better approximation by using a higher degree polynomial like a quadratic, since a parabola can "fit the curve" of y=f(x) better than a line can. the best quadratic approximationP(x) to f(x) near x-a must satisfy all of the following conditions:

P(a)=f(a)

P'(a)=f'(a)

P"(a)=f"(a)

Find the quadratic function P(x)=A+Bx+Cx² that is the best quadratic approximation to f(x)=cos x near x=0 [ You must solve for the values of the constants A,B,C that make the above three conditions true.]

Note that the linear approximation to f(x)=cos x near x=0 is just the constant function L(x)=1