Part a.
I'm guessing this is calculus. I know how to do it without calculus, but that's a bit more obscure.
y = 2x³ - ax² + bx + 2
We know (4,-46) is on the curve so y(4)=-46,
-46 = 2(4³) - a(4²) + b(4) + 2
0 = 128 +2 + 46 - 16a + 4b
0 = 176 - 16a + 4b
That's one equation. Now the derivative.
y' = 6x² - 2ax + b
We know (4,-46) is stationary so y'(4)=0.
0 = 6(4²) - 2a(4) + b = 96 - 8a + b
Now we have two equations in two unknowns. Doubling the last,
0 = 192 - 16a + 2b
Subtracting the first,
0 = (176-192) + 2b
b = 16/2 = 8
8a= 96 + b = 104
a = 104/8 = 13
Now we know
y = 2x³ - 13x² + 8x + 2
y' = 6x² - 26x + 8
y'' = 12x - 26
y''(4) = 12(4) - 26 = 48 - 26 = 22
It's positive, Concave Up Positive, it's a CUP, so a
Answer: MINIMUM
Part b.
y' = 6x² - 26x + 8
We already know one factor of y' is x-4 because y'(4)=0 so this is easy to factor:
y' = 0 = (x-4)(6x - 2)
6x=2
x = 1/3
y(1/3) = 2(1/3)³ - 13(1/3)² + 8(1/3) + 2 = 89/27
y''(1/3) = 12(1/3) - 26 = -22
Negative, concave down, a max.
Answer: (1/3, 89/27) is a MAXIMUM
