The cost to produce a product is modeled by the function f(x) = 3x2 − 6x + 11 where x is the number of products produced. complete the square to determine the minimum cost of producing this product.

Because this is a positive parabola that opens upwards, it natrually has a minimum value for the vertex. To complete the square we will first set the quadratic equal to 0 then move the constant over by subtraction. That gives us . One very important rule to follow about completing the square is that the leading coefficient ALWAYS be a +1. Our is a 3. So we will factor it out like this: . Next step is to take half the linear term, square it, and add it to both sides. Our linear term is 2. Half of 2 is 1, and 1 squared is 1. So we add 1 to both sides. BUT we cannot forget about that 3 we factored out. It refuses to be ignored. It is a multiplier. So when we add 1 into the set of parenthesis, we are REALLY adding in 3*1 = 3. We add 3 to the right side. All that looks like this: . The right side simplifies to -8, easy enough. What we did this for, this whole entire process, was to create a perfect square binomial on the left that will serve as the x coordinate of the vertex (actually the "h" since the vertex coordinates are (h, k)). Setting up that binomial looks like this now: . We move the -8 back over by addition, and find that the coordinates for the vertex are (1, 8). If x (h) is 1, that is "1 product" since x is products. If y (k) is 8, that is $8. What our vertex tells us in this particular situation is that it costs $8 to make 1 item.

with btw u said 100pts but it clearly says

i answerd and just got 10pts

no hate

required rate of return = 0.166 = 16.6%

step-by-step explanation:

under the capital asset pricing model (capm), the required or expected rate of return for a particular security or asset is given using the formula:

required rate of return = risk-free rate + beta*(market risk premium)

we are given that;

risk-free rate = 4% = 0.04

beta = 1.4

market risk premium = 9% = 0.09

we simply plug these values into the above formula;

required rate of return = 0.04 + 1.4*(0.09)

required rate of return = 0.04 + 0.126

required rate of return = 0.166 = 16.6%