Consider the following functions. f1(x) = 0, f2(x) = x, f3(x) = ex 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}.) {c1, c2, c3} = Incorrect: Your answer is incorrect. Determine whether f1, f2, f3 are linearly independent on the interval (−[infinity], [infinity]). linearly dependent linearly independent Correct: Your answer is correct.

a) Only a trivial solution exists.

b) Since there is only a trivial solution, then we infer that the set of functions is linearly independent.

a) According to Linear Algebra, a set of elements is linearly independent if and only if a coefficient at least within a linear combination under the following form is not zero:

(1)

Where:

If we know that , and , then we have the following expression:

Given that , we conclude that can be any real number and we can reduce (1) by Compatibility with Addition:

is a trascendental function bounded between 0 and and represents a polynomic function with all real numbers as domain. There is a trivial solution for , but nontrivial ones for all . Hence, we have the following finding:

, and have a set of the form for all .

Now if we consider and represents any real number, then we reduce (1) into this:

And the only choice to meet this linear combination is considering that .

Hence, we have a trivial solution.

b) In consequence, the three functions are linearly independent since it is guaranteed that trivial solution is the only existing solution.

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Linearly independent

Step-by-step explanation:

The given functions are:

We want to solve for and  so that on the interval .

Let us substitute to get:

This implies that:

--->eqn1

Let us differentiate each ter wrt x to get:

---->eqn2

Make the subject in equation (2) to get:

---->eqn3

Substitute into equation (1) to get:

Factor nicely to get:

Note that we are after constants that will this equation true for all x. The only way this could be possible is when

Put into eqn3 to get:

Let us substitute everything into g(x) to find

This implies that:

By the zero product principle , for r , we must have either or .

Since is true, is also true

Since  for all x, if , the functions are linearly independent.

you never say the total so we cant find the percentage used for petrol

step-by-step explanation:

50.6

step-by-step explanation: