Exercise 1.2.13

Find a formula for a cubic function f if f ( 1 ) = 6 and f ( − 1 ) = f ( 0 ) = f ( 2 ) = 0 .

Answers

Since f is a cubic function, we can find coefficients a , b , c , d ∈ ℝ such that

∀ ⁡ x ∈ ℝ : f ( x ) = a x 3 + b x 2 + cx + d .

Since f ( 1 ) = 6 , we have:

a + b + c + d = 6 . 		(1)

Since f ( − 1 ) = f ( 0 ) = f ( 2 ) = 0 , we have:

− a + b − c + d = d = 8 a + 4 b + 2 c + d = 0 .

From the equation above, we can conclude that d = 0 . From the equations − a + b − c = 0 and equation 1 we can write:

− a + b − c = − ( a + c ) + b = − ( 6 − b ) + b  [from  1    a + c = 6 − b ]  = − 6 + b + b = − 6 + 2 b = 0 .

2 b = 6 .

b = 3 .

Since 8 a + 4 b + 2 c = 0 and a + c = 6 − b = 6 − 3 = 3 , to find a and c we should solve the following system:

{ 8 a + 4 ⋅ 3 + 2 c = 0 a + c = 3 ⟺ { 8 a + 2 c = − 12 a + c = 3 ⟺ { 8 a + 2 c = − 12 2 a + 2 c = 6 ⟺ { 8 a + 2 c − ( 2 a + 2 c ) = − 12 − 6 2 a + 2 c = 6 ⟺ { 6 a = − 18 a + c = 3 ⟺ { a = − 3 c = 6 .

Thus, the formula of f is ∀ ⁡ x ∈ ℝ : f ( x ) = − 3 x 3 + 3 x 2 + 6 x .

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2023-08-05 07:03
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