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lana [24]
3 years ago
8

Consider the following functions. f1(x) = x, f2(x) = x2, f3(x) = 7x − 5x2 g(x) = c1f1(x) + c2f2(x) + c3f3(x) Solve for c1, c2, a

nd c3 so that g(x) = 0 on the interval (−[infinity], [infinity]). If a nontrivial solution exists, state it. (If only the trivial solution exists, enter the trivial solution {0, 0, 0}.)
Mathematics
1 answer:
Margaret [11]3 years ago
5 0

Answer:

Therefore the solution is = k{-7,5,1} where k ∈R

Step-by-step explanation:

Given that,

f₁(x) =x

f₂(x)= x²

f₃(x)= 7x - 5x²

Also,

g(x) = c₁f₁(x)+c₂f₂(x)+c₃f₃(x)

Putting the values of f₁(x), f₂(x) and f₃(x).

g(x) = c₁.x+c₂x²+c₃(7x-5x²)

Given condition that g(x)= 0

∴ c₁.x+c₂x²+c₃(7x-5x²)=0

⇒(c₁+7c₃)x +(c₂-5c₃)x² = 0

Comparing the coefficients of x and x²

∴c₁+7c₃=0               and       c₂-5c₃ =0

\Rightarrow c_1 =-7c_3                     \Rightarrow c_2=5c_3

Let c₃= k   [k∈R]  

Then c₁ = -7k   and   c₂=5k

Therefore the solution is = { c₁,c₂,c₃}  

                                           = {-7k, 5k, k}

                                            =k{-7,5,1}

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