Find the absolute minimum value on [0,positive infinity) for f(x)=(x+4)(x-2)squared

Question

absolute extrema

Answer 1

Hi Abbey:

 

not sure what the function is exactly, but the principle is as follow. 

 

It is a continuous function whose derivative exists.  Therefore, extreme values are either on the edge of the interval [0 , infinity], or at the zero slope points where the derivative is zero. 

 

So calculate the value of the function at x = 0;  then look at the value of the function when x increases to infinity (larger and larger values).  Looks like the larger x is, the larger the function, so it tends to + infinity as well.  Thankfully, we are looking for an absolute minimum, therefore the infinity value is out.

 

Then calculate the derivative, and find the x values for which it is zero.  Then calculate the value of the function for those zero slope x values, and compare to the value of the function at the edge, x = 0. 

 

Finally, pick the x value for which the function value is lowest.  That is your absolute minimum value.

 

Hope this helps!

 

  Jean-Michel