Question

1. Let f be the function that is defined for all real numbers x and that has the following properties.
(i) f''(x)=24x-18
(ii) f'(1)=-6
(iii) f'(2)=0
a. Find each x such that the line tangent to the graph of f at (x, f (x)) is horizontal
b. Write an expression for f (x)

Answer 1

f'(x) = 12x^2 - 18x + c
for f'(1) = -6
12 - 18 + c = -6
c = 0

so f'(x) = 12x^2 - 18x

Then you say f'(2) = 0 , which does not verify with what I have so far.
Did you mean f(2) = 0 ?
I will assume you meant that.
f(x) = 4x^3 - 9x^2 + k
given f(2) = 0
32 - 36 + k = 0
k = 4

f(x) = 4x^3 - 9x^2 + 4

leaving up to you: .....

a) .......
tell me what you would do.

Answer 2

f''(x) = 24x - 18

f'(x) = 12x^2 - 18x + C
Since f'(1) = -6, then
-6 = 12(1)^2 - 18(1) + C
-6 = 12 - 18 + C
-6 = -6 + C
C = 0

So f'(x) = 12x^2 - 18x

f(x) = 4x^3 - 9x^2 + C
Since f(2) = 0, then
0 = 4(2)^3 - 9(2)^2 + C
0 = 4(8) - 9(4) + C
0 = 32 - 36 + C
0 = -4 + C
4 = C

f(x) = 4x^3 - 9x^2 + 4

For Part A, you know that f'(x) = 12x^2 - 18x
f'(x) = 0
12x^2 - 18x = 0
6x(2x-3) = 0
x = 0 or x = 3/2
Just evaluate f(0) and f(3/2) to find the y-coordinates.