Let f(x) be a differentiable function, defined for all real numbers x, with the following properties. Use all three properties to find f(x). Show work.
f'(x)=ax^2+bx
f'(1)=6 and f''(1)=18
2
\(\int\)f(x)dx=36
1
P.S. sorry for the poor formatting of the third property I'm not too tech savvy.
+128407 f ' ( x) = ax^2 + bx
f ' (1) = 6
a(1)^2 + b(1) = 6
a + b = 6 ⇒ b = 6 - a (1)
f " (x) = 2ax + b
f " (1) =18
2a(1) + b =18
2a + b = 18 (2)
Sub (1) into (2) for b
2a + 6 - a = 18
a + 6 = 18
a = 12
b = 6 - 12 = - 6
So
f'x) = 12x^2 - 6x
f(x) = 4x^3 - 3x^2 + C
So
2
∫ 4x^3 - 3x^2 + C dx = 36
1
2
[ x^4 - x^3 + Cx ] = 36
1
[2^4 - 2^3 + 2C ] - [ 1^4 - 1^3 + C ] = 36
[ 16 - 8 + 2C ] - [ C] = 36
8 + C = 36
C = 28
So
f(x) = 4x^3 - 3x^2 + 28
