Question

If , show that exists for all real x , and find it.

Answer 1

Let, f( x )= | x | 3 .

The function | x |is defined as follows,

| x |={ x,      x≥0 −x,    x<0 }

The value of f( x )for x≥0 is f( x )= x 3 .

Differentiate both sides with respect to x.

f( x )= x 3 f ′ ( x )=3 x 2

Again, differentiate both sides with respect to x.

f ″ ( x )=6x

The value of f( x )for x<0 is f( x )=− x 3 .

Differentiate both sides with respect to x.

f( x )= ( −x ) 3 f ′ ( x )=−3 x 2

Again, differentiate both sides with respect to x.

f ″ ( x )=−6x

Thus, the value of f ″ ( x )exists for all real value and it is given by,

f ″ ( x )={ 6x,x≥0 −6x,x<0 }