Answer:
a = 2, b = -2
P is at a minimum
Explanation:
For any polynomial function, the turning point is either a maximum or a minimum
It can be determined by taking the first derivative of the function and setting it equal to 0 and solving for x and y
In this case we are given the turning point as x=1, y = 3 and we have to calculate a and b in the equation y = ax² + (ab)x + 5
The first derivative of y, y' = 2ax + ab
If we set this equal to 0 we get 2ax + ab = 0 ==> 2ax = -ab ==> b = -2
So the equation is of the form y = ax² -2ax + 5 (subbing for b)
Since we know that at x = 1, y =3 substitute y and x values in the above equation and solve for a
y = 3 = a(1²) - 2a(1) + 5
a - 2a + 5 = 3
-a + 5 = 3
a = 2
So the equation is of the form y = 2x² -4x + 5
If we plot this we will find that P is a minimum point
However, we can always determine mathematically if P is a max or min by taking the second derivative of the original function and noting the sign. If it is positive, the point is a minimum, and if it is negative, the point is a maximum.
Taking derivatives,
y' = 4x - 4
y'' = (4x-4)' = 4
The sign is positive so P is a minimum
Graph attached for reference