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PLEASE HELP “For the quartic function f(x)=ax^4 +bx^2 +cx + d find the values of a, b, c and d. There is a maximum at (0, -6) and a minimum at (1, -8).

User Kassim
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Answer:

(a, b, c, d) = (2, -4, 0, -6)

Explanation:

You want the coefficients in f(x) = ax^4 +bx^2 +cx +d such that (0, -6) is a maximum and (1, -8) is a minimum.

Function values

The given points help you write two equations in the four unknown values.

f(0) = -6

a·0 +b·0 +c·0 +d = -6 ⇒ d = -6

f(1) = -8

a·1 +b·1 +c·1 -6 = -8 ⇒ a +b +c = -2

Derivative values

The derivative of the function is ...

f'(x) = 4ax^3 +2bx +c

At the given extremes, the derivative is zero, so we have ...

4a·0 +2b·0 +c = 0 ⇒ c = 0

4a·1 +2b·1 +0 = 0 ⇒ 2a +b = 0

Solution

Using the equations we have, we can solve for 'a' and 'b':

  • a + b = -2
  • 2a + b = 0

Subtracting the first equation from the second, we get ...

(2a +b) -(a +b) = (0) -(-2)

a = 2

The first equation gives the value of b:

b = -2 -a = -2 -2

b = -4

The values of a, b, c, d are ...

(a, b, c, d) = (2, -4, 0, -6)

PLEASE HELP “For the quartic function f(x)=ax^4 +bx^2 +cx + d find the values of a-example-1
User Imgnx
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