Question

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, and 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}.)

Answer 1

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}

Answer 2

The required trivial solution set is $\{(-7k,5k,k);k\ \epsilon \mathbb{R}}\}$

Trivial solution:

Trivial solutions are the solutions to some equations which have a simple structure.

Given that,

$f_1(x)=x,f_2(x)=x^2,f_3(x)=7x-5x^2$

$g(x)=c_1f_1(x)+c_2f_2(x)+c_3f_3(x)\\=c_1x+c_2x^2+c_3(7x-5x^2)\\=(c_1+7c_3)x+(c_2-5c_3)x^2\\g(x)=0\\c_1+7c_3=0\\c_2-5c_3=0$

Let $c_3=k$ then $c_1=-7k,c_2=5k$

Hence, the solution set is,

$\{(-7k,5k,k);k\ \epsilon \mathbb{R}}\}$

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