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.