Answer to Question #203205 in Real Analysis for Rajkumar

Question

Check whether the function f given by:

f (x) = (x − 4)³ (x +1)²

has local maxima and local minima.

Answer 1

f= (x − 4)³ (x +1)²

f'=2(x − 4)³ (x +1)+3(x − 4)² (x +1)²

f''=2(x − 4)³ +6(x − 4)² (x +1)+6(x − 4)² (x +1)+6(x − 4)(x +1)²

Now, the critical points are,

f'=0

2(x − 4)³ (x +1)+3(x − 4)² (x +1)²=0

(x − 4)²(x +1) [2(x-4)+3(x +1)]=0

(x − 4)²(x +1) (5x -5)=0

x=4,4,-1,1

Then,

f''(4)=0

f''(-1)=-250<0 (increasing)

f''(1)=90>0 (decreasing)

Thus,  local maxima at x=1 and local minima at x=-1.