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(1 point) Find the values of constants a,b, and c so that the graph of y= bx+c

x 2
+a

has a local minimum at x=8 and a local maximum at (−1,−5). a
(−1

a=
b=
c=


User The
by
8.1k points

1 Answer

2 votes

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.

User Fabian Jakobs
by
8.7k points

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