Question
Find the values of a , b , and c in the quadratic function p(x)=ax^2+bx+c such that p(2)=6, pŒ(2)=2, and pŒŒ(2)=3.
Answers
MathMate
f(x)=ax²+bx+c
f'(x)=2ax+b
f"(x)=2a
f"(2)=3 => 2a=3 => a=3/2
f'(2)=2 => 2a(2)+b=2 => 3(2)+b=2 => b=-4
f(2)=6 => a(2)²+b(2)+c=6
=> 4a+2b+c=6
=> 6-8+c=6
=> c=8
Therefore
f(x)=(3/2)x²-4x+8
Using this final definition of f(x), verify that f(2)=6, f'(2)=2, f"(2)=3.
f'(x)=2ax+b
f"(x)=2a
f"(2)=3 => 2a=3 => a=3/2
f'(2)=2 => 2a(2)+b=2 => 3(2)+b=2 => b=-4
f(2)=6 => a(2)²+b(2)+c=6
=> 4a+2b+c=6
=> 6-8+c=6
=> c=8
Therefore
f(x)=(3/2)x²-4x+8
Using this final definition of f(x), verify that f(2)=6, f'(2)=2, f"(2)=3.