Local maxima, minima

Question

!!f(x)={ [2(x+3)^3, -4 < x <= -2],[x^(2/3),-2 < x < 2]:}!!

The total number of local maxima and local minima are:

  • !!0!!
  • !!1!!
  • !!2!!
  • !!3!!

Solution

So we will need to find the points such that on either sides of the point !!x!!, !!f(y) < f(x)!! if !!x!! is local maxima and !!f(y) > f(x)!! if !!x!! is local minima for all !!y!! near !!x!!.

We first consider the function !!2(x+3)^3!!

Now this is a continuos function and the derivative of the function is zero at !!x=-3!!. But on either sides of this point the slope of function is positive so !!x=-3!! is neither a local maxima nor a local minima.

More over the slope is always positive so there will be no local maxima for !!-4 < x <-2!!.

Now we consider the point where the 2 functions meet which is !!x=-2!!. Now for !!x<-2!! !!f'(x)>0!! but for !!x > -2!! !!f'(x)=2/3 x^(-1/3)<0!!.

So !!x=-2!! is a local maxima.

The second function !!f(x)=x^2/3!! also attains local minima at !!x=0!! because for !!y<0!! but near 0, !!f'(y)<0!! and !!f'(y)>0!! for !!y>0!! near 0.

So there are 2 solutions.

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