NDA2017MathematicsApplication of DerivativesActual
Which one of the following is correct in respect of the function f( x )= x ( x -1)( x +1) ?
Options
- AThe local maximum value is larger than local minimum value
- BThe local maximum value is smaller than local minimum value
- CThe function has no local maximum
- DThe function has no local minimum
Correct answer
A. The local maximum value is larger than local minimum value
Step-by-step solution
f(x)=x(x-1)(x+1)=x (x^2-1 ) Differentiating, we get f^ (x)=x(2 x)+ (x^2-1 )=2 x^2+x^2-1=3 x^2-1 Again differentiating f^ (x)=6 x At ^ (x)=0 3 x^2-1=0 x^2= 1 3 x= 1 3 Maximum value of f(x)=f ( -1 3 )= -1 3 ( ( -1 3 )^2-1 )= -1 3 ( 1 3 -1 )= -1 3 ( -2 3 )= 2 3 3 Minimum value of f(x)=f ( 1 3 )= 1 3 ( 1 3 -1 )= -2 3 3