Answer: f(x) = 2*x^2 + 4*x + 2.
Step-by-step explanation:
Ok, we have that:
f''(x) = 4.
Then, integrating, we have that:
f'(x) = 4*x + a.
and from the conditions we know that:
f'(2) = 12 = 4*2 + a = 8 + a
12 = 8 + a
a = 12 - 8 = 4
then f'(x) = 4*x + 4.
now we can integrate again and get:
f(x) = (1/2)*4*x^2 + 4*x + b.
f(x) = 2*x^2 + 4x + b
and we know that f(2) = 18:
f(2) = 18 = 2*2^2 + 4*2 + b = 8 + 8 + b
18 = 16 + b
18 - 16 = b = 2.
Then the function is:
f(x) = 2*x^2 + 4*x + 2.
