AP EAMCET202123 Aug 2021Morning ShiftMathematicsApplication of DerivativesActual

The absolute minimum value of x^4-x^2-2 x+5 is

Options

Aequal to 5
Bequal to 3
Cequal to 7
DDoes not exist

Correct answer

B. equal to 3

Step-by-step solution

x^4-x^2-2 x+5=f(x) (say) Then, f^ (x)=4 x^3-2 x-2 Equate f^ (x)=0 4 x^3-2 x-2=0 2 x^3-x-1=0 2 x (x^2-1 )+(x-1)=0 2 x(x-1)(x+1)+(x-1)=0 (x-1)[2 x(x+1)+1]=0 (x-1) (2 x^2+2 x+1 )=0 x-1=0 and 2 x^2+2 x+1=0 x=1 and x=- 2 4-8 4 (Imaginary) x=1 and x=(-1 1 i) / 2f^ (x)=12 x^2-2f^ (x) / x=1=10>0 f(x) is minimum at x=1 Now, f( l )=( l )^4-( l )^2-2+5=3 Absolute minimum value is 3 .

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