Find a second-degree polynomial P such that P(3) = 19, P'(3) = 14, and P''(3) = 6.

Question

Find a second-degree polynomial P such that P(3) = 19, P'(3) = 14,

 and P''(3) = 6.

P(X)=______

Answer 1

A second degree polynomial can be written as f(x) = ax² + bx + c. Then the first and second derivatives are:

f '(x) = 2ax + b

f ''(x) = 2a

So now we can plug in the data points.

For f(x): f(3) = 19 = a(3)² + b(3) + c or:

9a + 3b + c = 19

For f '(x): f '(3) = 14 = 2a(3) + b or:

6a + b = 14

For f ''(x): f ''(3) = 6 = 2a or

2a = 6

a = 3

Back substituting into the second equation:

6(3) + b = 14

b = -4

Back substituting into the first equation:

9(3) + 3(-4) + c = 19

c = 4

So f(x) = 3x² - 4x + 4