question_answer

Let \[f:[-1,3]\to [-8,72]\] be denned as \[f(x)=4{{x}^{3}}-12x,\] then f is

A) Injective but not surjective    

B)                        Bijective

C) Neither injective nor surjective                         

D) Surjective but not injective

Correct Answer: D

Solution :

[d] \[f(x)=4{{x}^{3}}-12x\] \[\therefore \,\,\,\,\,\,f'(x)=12({{x}^{2}}-1)=12(x+1)(x-1)\] Sign scheme of \[f'(x)\] is as shown in the following figure: Thus, \[x=-1\]is point of maxima and \[x=1\]is point minima. The graph of the function \[f(x)=4{{x}^{3}}-12x\]for \[x\in R\]is as shown in the figure: For \[x\in R[-1,3],\] function \[f(x)\]is surjective but not injective.