Calculus
posted by Leanna on .
A cubic polynomial function f is defined by f(x) = 4x^3 + ax^2 + bx + k?
A cubic polynomial function f is defined by:
f(x) = 4x^3 + ax^2 + bx + k
where a, b, and k are constants.
The function f has a local minimum at x= 1, and the graph of f has a point of inflection at x= 2.
a) Find the values of a and b
b) If ∫ (from 0 to 1) f(x) dx = 32, what is the value of k?
I found a=24 and b=36
i don't know about B)

From a=24,b=36,
we find
f'(x)=0 has roots at x=3 and 1, so that checks.
f"(x)=0 has a root at x=2, so that checks too.
To solve B, we integrate
f(x):=4*x^3+24*x^2+36*x+k
from 0 to 1:
I=∫f(x)dx
=x^4+8*x^3+18*x^2+k*x
Evaluate I from 0 to 1 to give k+27
But since is given as I=32, so
I=32=k+27
Solve for k.