SOLUTION: What is the optimal value of this function: y=0.5x^2+2x+3

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Question 253264: What is the optimal value of this function: y=0.5x^2+2x+3
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'm assuming that when you say 'optimal', you mean smallest (as in say the lowest cost).


Take note that y=0.5x%5E2%2B2x%2B3 is equivalent to y=%281%2F2%29x%5E2%2B2x%2B3 since 1%2F2=0.5


To find the optimal value, we need to find the vertex.


In order to find the vertex, we first need to find the x-coordinate of the vertex.


To find the x-coordinate of the vertex, use this formula: x=%28-b%29%2F%282a%29.


x=%28-b%29%2F%282a%29 Start with the given formula.


From y=%281%2F2%29x%5E2%2B2x%2B3, we can see that a=1%2F2, b=2, and c=3.


x=%28-%282%29%29%2F%282%281%2F2%29%29 Plug in a=1%2F2 and b=2.


x=%28-2%29%2F%281%29 Multiply 2 and 1%2F2 to get 1.


x=-2 Divide.


So the x-coordinate of the vertex is x=-2. Note: this means that the axis of symmetry is also x=-2.


Now that we know the x-coordinate of the vertex, we can use it to find the y-coordinate of the vertex.


y=%281%2F2%29x%5E2%2B2x%2B3 Start with the given equation.


y=%281%2F2%29%28-2%29%5E2%2B2%28-2%29%2B3 Plug in x=-2.


y=%281%2F2%29%284%29%2B2%28-2%29%2B3 Square -2 to get 4.


y=2%2B2%28-2%29%2B3 Multiply 1%2F2 and 4 to get 2.


y=2-4%2B3 Multiply 2 and -2 to get -4.


y=1 Combine like terms.


So the y-coordinate of the vertex is y=1.


So the vertex is .


Since the max/min occurs at the vertex as the 'y' value, this means that the min is y=1. So the optimal value is is y=1.


Here's a graph to visually confirm the answer:




Graph of y=0.5x%5E2%2B2x%2B3