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March 18th, 2017, 09:45 PM  #1 
Member Joined: Jan 2017 From: US Posts: 91 Thanks: 5  How to Find a Minimum/Maximum in a Function?
What are the minimums and maximums in a function, and what are their purpose? How would you pick them out of a function? Here is a function if you want to use it as an example: f(x)=3x$\displaystyle ^2$+1x 
March 18th, 2017, 09:50 PM  #2 
Member Joined: Mar 2017 From: Tasmania Posts: 36 Thanks: 2 
You can complete the square to find the min and max but I believe there is also a formula you can use

March 18th, 2017, 10:24 PM  #3  
Senior Member Joined: Sep 2015 From: Southern California, USA Posts: 1,413 Thanks: 717  Quote:
You can tell whether it is facing upwards or downwards by looking at the sign of the coefficient of the x^2 term. A positive coefficient means the parabola faces upwards. Negative means it faces downwards. So if we compete the square and put the function into vertex form we can a) pick the vertex out, and you know this will be either the min or max b) examine the coefficient of $(xh)^2$ to detemine it's sign and whether the point in (a) is a min or a max. $f(x) = 3x^2  x +1 =$ $3\left(x^2 + \dfrac x 3  \dfrac 1 3\right)=$ $3\left(\left(x+\dfrac 1 6\right)  \left(\dfrac 1 6 \right)^2  \dfrac 1 3\right) = $ $3\left(\left(x+\dfrac 1 6\right)^2 \dfrac {13}{36}\right) =$ $3\left(x+\dfrac 1 6\right)^2 + \dfrac{13}{12}$ We see that the $(xh)^2$ term has coefficient $3$ and thus the vertex, $\left(\dfrac 1 6,\dfrac {13}{12}\right)$ is a maximum.  
March 19th, 2017, 12:17 AM  #4  
Member Joined: Jan 2017 From: US Posts: 91 Thanks: 5  Quote:
 
March 31st, 2017, 01:40 AM  #5  
Newbie Joined: Mar 2017 From: Portland, OR Posts: 5 Thanks: 0  Quote:
You could also graph this equation if you have something like a Ti89 and hit F5 and select either minimum (if pointing up) or maximum (if pointing down) then go to the left side (lower bound) press enter and right side of the curvature (upper bound) (or go right then left, doesn't matter), hit Enter and boom there it is. You could find the vertex a number of ways. Firstly, find x=b/2a and plug that x value in to get y afterwards and that point is your vertex. Or you could put the equation in vertex form by completing the square.  

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