Local extrema Use the sign pattern for the derivative

df/dx=6(x-1)(x-2)^2(x-3)^3(x-4)^4

Dear student

 

If we use wavy curve method to check the sign of df/dx

Then we see that in

-infinity to 1 , it is + sign

And 1 to 3 it is negative

Again 3 to infinity it is positive

 

Hence local maximum exist at x = 1

And local minimum exist at x = 3

 

Hope it helps

Dear student 

Since the second derivative is zero, the function is neither concave up nor concave down at x = 0. It could be still be a local maximum or a local minimum and it even could be an inflection point. Let's test to see if it is an inflection point. We need to verify that the concavity is different on either side of x = 0.

Good Luck