Maths · Limits, continuity and differentiability
Assertion If the series represented by function =\boldsymbol{x}^{2}+\boldsymbol{
Assertion If the series represented by function \( \boldsymbol{f}(\boldsymbol{x})=\boldsymbol{x}^{2}+\boldsymbol{x}^{4}+\boldsymbol{x}^{\boldsymbol{6}}+\boldsymbol{x}^{\boldsymbol{8}}+\ldots \) converges, then function \( \boldsymbol{g}(\boldsymbol{x})=[\boldsymbol{x}] \) (where [.] denotes the greatest integer function) is continuous at one fixed point of \( \boldsymbol{f}(\boldsymbol{x}) \) Reason \( \boldsymbol{f}(\boldsymbol{x})=\boldsymbol{x} \Rightarrow \boldsymbol{x}^{2}+\boldsymbol{x}-\mathbf{1}=\mathbf{0} \) which gives two fixed points.
- A. If both assertion and reason are correct and reason is the correct explanation of the assertion
- B. If both assertion and reason are correct but reason is not correct explanation of the assertion
- C. If assertion is correct, but reason is incorrect
- D. If assertion is incorrect, but reason is correct
Step-by-step solution
The series f(x)=x^2+x^4+... converges for |x|<1, with sum x^2/(1-x^2). Fixed points satisfy x^2/(1-x^2)=x, giving x(x^2+x-1)=0. Only x=0 and x=(√5-1)/2 ≈0.618 lie in (-1,1). g(x)=[x] is continuous at x≈0.618 (not an integer) but discontinuous at x=0. Thus assertion is correct. The reason correctly gives x^2+x-1=0 (from nonzero fixed points), yielding two fixed points, but does not explain continuity; hence it is correct but not the correct explanation.
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