Answer:
Therefore, the value of a is 0, the value of b is 0, and the value of c can be any real number since it does not affect the local minimum and maximum.
Explanation:
To find the values of constants a, b, and c, we need to consider the properties of local minimum and maximum.
First, let's consider the local minimum at x=8. At a local minimum, the derivative of the function is equal to zero and the second derivative is positive.
Taking the derivative of y=bx^2+ax+c with respect to x, we get:
dy/dx = 2bx + a
Setting this equal to zero, we have:
2bx + a = 0
Since this needs to hold true at x=8, we can plug in 8 for x:
2b(8) + a = 0
16b + a = 0
This gives us the first equation: 16b + a = 0.
Now, let's consider the local maximum at (-1, -5). At a local maximum, the derivative of the function is equal to zero and the second derivative is negative.
Taking the derivative of y=bx^2+ax+c with respect to x again, we get:
d^2y/dx^2 = 2b
Setting this equal to zero, we have:
2b = 0
Since this needs to hold true at x=-1, we can plug in -1 for x:
2b = 0
This gives us the second equation: 2b = 0.
Solving the system of equations:
16b + a = 0 (equation 1)
2b = 0 (equation 2)
From equation 2, we can see that b must be equal to 0.
Substituting b = 0 into equation 1:
16(0) + a = 0
a = 0
Therefore, the value of a is 0, the value of b is 0, and the value of c can be any real number since it does not affect the local minimum and maximum.