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Difference between revisions of "Optimization"

(→‎Process)
(Expanded the scope of optimization to include more than just quadratics.)
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The '''optimization''' of a [[quadratic equation]] is the process to find the [[maximum]] or [[minimum]] of said quadratic.  
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'''Optimization''' is simply finding the [[maximum]] or [[minimum]] possible value.  In order to prove that a value is a maximum or minimum, one needs to prove that the value is attainable and that there is no higher or lower value (depending on the problem) that works.
  
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==Process==
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==Optimization Techniques==
  
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It involves [[conversion|converting]] a quadratic to the [[standard form]] <math>a(x-h)^2+k</math> by [[completing the square]]. Then by the [[Trivial Inequality]],  the maximum or minimum (it depends on which way the [[graph]] of the quadratic is facing) is <math>k</math>.
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* There are multiple ways to determine the maximum or minimum (depending of the leading term) of a quadratic (depending on the form).
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** If the quadratic is in the form <math>a(x-h)^2+k</math> (vertex form), the maximum or minimum of the quadratic is <math>k</math> by the [[Trivial Inequality]].
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** If the quadratic is in the form <math>ax^2 + bx + c</math> (standard form), the maximum or minimum of the quadratic is achieved when <math>x = -\tfrac{b}{2a}</math>.  This can be derived by [[completing the square]].
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* The maximum of <math>\sin (x)</math> and <math>\cos (x)</math> is 1, and the minimum of <math>\sin (x)</math> and <math>\cos (x)</math> is -1.
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* One can also use [[coordinate geometry]] to determine the maximum or minimum.  Optimization is often done when two figures touch each other exactly once.
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* In [[calculus]], for a function <math>f(x)</math>, the local maximums and local minimums are part of the critical points of the function.  The x-values of the critical points can be found by taking the derivative of <math>f(x)</math> and setting it to equal 0.  In order to find the absolute maximum or minimum, one needs to also check the endpoints of an interval.
  
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==="Formula"===
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[[Category: Algebra]]
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[[Category: Calculus]]
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To optimize a quadratic, one might use the method described above, or one could use this other, smoother, method:
 
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If <math>a>0</math>, then the quadratic <math>ax^2+bx+c=0</math> reaches its minimum when <math>x=-\frac{b}{2a}</math>, while when <math>a<0</math>, the quadratic reaches its ''maximum'' when <math>x=-\frac{b}{2a}</math>.
 
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===Alternative===
 
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The alternative requires [[calculus]]. The optimum point has a [[tangent]] line with a slope of 0. Thus, calculating the [[derivative]] and setting it equal to 0 will also give an answer. Assuming that the original [[quadratic]] is <math>ax^2+bx+c</math>, the derivative is <math>2ax+b</math>. Setting <math>2ax+b = 0</math>, we find that <math>x = -\frac{b}{2a}</math> .
 

Revision as of 16:27, 6 June 2019

Optimization is simply finding the maximum or minimum possible value. In order to prove that a value is a maximum or minimum, one needs to prove that the value is attainable and that there is no higher or lower value (depending on the problem) that works.

Optimization Techniques

  • There are multiple ways to determine the maximum or minimum (depending of the leading term) of a quadratic (depending on the form).
    • If the quadratic is in the form $a(x-h)^2+k$ (vertex form), the maximum or minimum of the quadratic is $k$ by the Trivial Inequality.
    • If the quadratic is in the form $ax^2 + bx + c$ (standard form), the maximum or minimum of the quadratic is achieved when $x = -\tfrac{b}{2a}$. This can be derived by completing the square.
  • The maximum of $\sin (x)$ and $\cos (x)$ is 1, and the minimum of $\sin (x)$ and $\cos (x)$ is -1.
  • One can also use coordinate geometry to determine the maximum or minimum. Optimization is often done when two figures touch each other exactly once.
  • In calculus, for a function $f(x)$, the local maximums and local minimums are part of the critical points of the function. The x-values of the critical points can be found by taking the derivative of $f(x)$ and setting it to equal 0. In order to find the absolute maximum or minimum, one needs to also check the endpoints of an interval.
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