Answer:
a = 6 and b = -1.
Explanation:
To find the maximum value of y = -x²+ ax + b we differentiate it with respect to x to find the value of x at which it is maximum.
So dy/dx = d(-x²+ ax + b)/dx = -2x + a
we equate it to zero.
-2x + a = 0
-2x = -a
a = 2x
Now, given that x = 3 at the maximum value,
a = 2x = 2(3) = 6
Also, substituting y = 8 and x = 3 into y, we have
y = -x²+ ax + b
8 = -(3)²+ a(3) + b
8 = -9 + 3a + b
3a + b = 8 + 9
3a + b = 17
So, b = 17 - 3a
substituting a = 6, we have
b = 17 - 3(6)
b = 17 - 18
b = -1
So, a = 6 and b = -1.