Difference between revisions of "Calc2.52"

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To find the local minimums and maximums of a function, find the critical points by finding the derivative and finding the points where it is equal to 0 or undefined.  If the original function is defined at these points, then these are critical points.
 
To find the local minimums and maximums of a function, find the critical points by finding the derivative and finding the points where it is equal to 0 or undefined.  If the original function is defined at these points, then these are critical points.
 
<br><br>
 
<br><br>
<math>f'(x)=\frac{1}{2\sqrt{x}}\</math>
+
<math>f'(x)=\frac{1}{2\sqrt{x}}\,</math>
 
<br><br>
 
<br><br>
 
This function is never equal to 0 as the numerator is never equal to 0.  It is undefined at <math>x=0</math>.  This is our only critical point.  However, the square root function is only defined for nonnegative real numbers so <math>f(x)</math> is not defined to the left of <math>x=0</math>.  Thus, it can not have a local minimum or maximum at <math>x=0</math>.  <math>x=0</math> does give us the lowest point on the entire square root function as it starts at <math>0</math> and increases thereafter.
 
This function is never equal to 0 as the numerator is never equal to 0.  It is undefined at <math>x=0</math>.  This is our only critical point.  However, the square root function is only defined for nonnegative real numbers so <math>f(x)</math> is not defined to the left of <math>x=0</math>.  Thus, it can not have a local minimum or maximum at <math>x=0</math>.  <math>x=0</math> does give us the lowest point on the entire square root function as it starts at <math>0</math> and increases thereafter.

Latest revision as of 02:53, 8 March 2006

Find the local minimums and maximums of the function f(x)={\sqrt  {x}}.

To find the local minimums and maximums of a function, find the critical points by finding the derivative and finding the points where it is equal to 0 or undefined. If the original function is defined at these points, then these are critical points.

f'(x)={\frac  {1}{2{\sqrt  {x}}}}\,

This function is never equal to 0 as the numerator is never equal to 0. It is undefined at x=0. This is our only critical point. However, the square root function is only defined for nonnegative real numbers so f(x) is not defined to the left of x=0. Thus, it can not have a local minimum or maximum at x=0. x=0 does give us the lowest point on the entire square root function as it starts at 0 and increases thereafter.


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