Minimizing related functions Complete each of the following ...

Question

Minimizing related functions Complete each of the following parts.

b. What value of x minimizes ƒ(x) = (x- a₁)² + (x - a₂)² + (x - a₃)² , for constants a₁, a₂, and a₃?

Accepted Answer

Welcome back everyone. Determine the value of x that minimizes F of X equals x minus V2 + x minus 12 + x + 52. We're given 4 answer choices A says -4/3, B 1/3, C 2/3 and D2. So essentially if we want to perform minimization, we have to identify the derivative of F of X. So we're differentiating x minus V squad, that is going to be our first derivative, then we're going to identify the derivative of X minus 1 squad. And we are going to find the derivative of X + 5 squad. So here we have composite functions. We have to apply the chain rule. First rule, let's apply the power rule for the first time we're going to get 2 multiplied by X minus 3. And now according to the chain rule we're multiplying by the derivative of X minus 3 and we continue. In the same fashion. So, for the second term, we're going to get 2 multiplied by X minus 1, multiplied by X minus 1 derivative. And then for the final term, we're going to get 2 multiplied by X + 5, multiplied by X + 5 derivative. Notice that each derivative is equal to 1 because the derivative of X is simply equal to. 1 and the derivative of a constant is 0, right? We can cross out all of the remaining derivatives and make them equal to 1. So overall we Going to factor out 2, which gives us X minus 3 plus X minus 1. Plus X + 5. And this is equal to two imprecies, we combine the like terms, so 3 X minus 3 minus 1 + 5, right? So this is -4 + 5, we're simply adding 1. So we have our derivative which is 6 x + 2. And basically what we have to do is set it equal to 0 because we want to identify the critical points. So if 6 X + 2 is equal to 0. Then we are subtracting 2 from both sides and dividing by 6. This gives us X equals -13 as the critical point. And then we want to prove that this critical point is a point of a minimum. So we want to apply the second derivative test. Let's identify the second derivative. This is the derivative of 6 x + 2. Which gives us 6. Now what does that mean? Well, it's positive for any value of X, including the critical point, meaning the the function is concave up. And if a function is concave up, we indeed have a local minimum at the critical point. Therefore, our final answer to this problem corresponds to the answer choice B-1/3. Thank you for watching.

M​​inimizing related functions Complete each of the following parts.b. What value of x minimizes ƒ(x) = (x- a₁)² + (x - a₂)² + (x - a₃)² , for constants a₁, a₂, and a₃?

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Minimizing related functions Complete each of the following parts.
b. What value of x minimizes ƒ(x) = (x- a₁)² + (x - a₂)² + (x - a₃)² , for constants a₁, a₂, and a₃?