Question

x³-3x+8 increasing & decreasing​

Answer 1

f(x)=x3−3x+2.  

We want to find out the intervals in which it is increasing and the intervals in which it is decreasing.

f′(x)=3x2−3=3(x2−1)=3(x+1)(x−1).

The function f(x) is increasing in the regions where f′(x)>0.

f′(x)>0 when (x+1)>0 and (x−1)>0 or when (x+1)<0 and (x−1)<0.

i.e. when x>−1 and x>1 or when x<−1 and x<1.

i.e. when x>1 or when x<−1.

⇒ The function f(x) is increasing in the interval R−[−1,1].

The function f(x) is decreasing in the regions where f′(x)<0.

f′(x)<0 when (x+1)>0 and (x−1)<0 or when (x+1)<0 and (x−1)>0.

i.e. when x>−1 and x<1 or when x<−1 and x>1.

i.e. when x>−1 and x<1.

⇒ The function f(x) is decreasing in the interval (−1,1).

When x=−1 or x=1,f′(x)=0 and these points are inflection points.

@GAVYA