Most Important Selected Qs for JEE AdvancedMathematicsApplication of Derivatives
Paragraph: If f: R R and f(x)=g(x)+h(x) where g(x) is a polynomial and h(x) is a continuous and differentiable bounded function on both sides, then f(x) is onto if g(x) is of odd degree and f(x) is into if g(x) is of even degree. To check whether f(x) is one-one we need to differentiate f(x) . If f^ (x) changes sign in domain of f then f is many-one else one-one Question: f: R R and f(x)= x (x^4+1 )(x+1)+x^4+2 x^2+x+
Options
- Aone-one into
- Bmany-one onto
- Cone-one onto
- Dmany-one into
Correct answer
D. many-one into
Step-by-step solution
aligned & f(x)=x^4+1+ 1 x^2+x+1 = even degree polynomial + bounded function 1 x^2+x+1 (0, 4 3 ) & f^ (x)= 4 x^3 (x^2+x+1 )^2-2 x-1 (x^2+x+1 )^2 aligned f^ (x)=0 has atleast one root which is repeated odd number of times or it has one root which is not repeated since numerator of f^ (x) is a polynomial of degree 7 . f(x)=0 has a point of extrema.