Question
The global minimum value of $$f\left( x \right) = {x^4} - {x^2} - 2x + 6$$ is :
A.
6
B.
8
C.
4
D.
nonexistent
Answer :
4
Solution :
$$\eqalign{
& f'\left( x \right) = 4{x^3} - 2x - 2 \cr
& \therefore \,f'\left( x \right) = 0 \cr
& \Rightarrow 2{x^3} - x - 1 = 0 \cr
& \Rightarrow \left( {x - 1} \right)\left( {2{x^2} + 2x + 1} \right) = 0 \cr
& \therefore \,x = 1 \cr
& {\text{Now, }}f''\left( x \right) = 12{x^2} - 2\,\,\,\,\,\,\,\,\,\therefore f''\left( 1 \right) = 12 - 2 > 0 \cr
& {\text{Hence, global min}}f\left( x \right) = 1 - 1 - 2 + 6 = 4 \cr} $$