Question

If the maximum value of y =-x²+ax+b is 8 when x = 3, find the values of a and b​

Answer 1

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.