jee-advanced 2009 Q26

jee-advanced · India · paper2 Differentiating Transcendental Functions Evaluate derivative at a point or find tangent slope

For the function $$f(x)=x\cos\frac{1}{x},\quad x\geq1,$$ (A) for at least one $x$ in the interval $[1,\infty),f(x+2)-f(x)<2$
(B) $\lim_{x\rightarrow\infty}f^{\prime}(x)=1$
(C) for all $x$ in the interval $[1,\infty),f(x+2)-f(x)>2$
(D) $f^{\prime}(x)$ is strictly decreasing in the interval $[1,\infty)$

(B), (C), (D)

For the function
$$f(x)=x\cos\frac{1}{x},\quad x\geq1,$$
(A) for at least one $x$ in the interval $[1,\infty),f(x+2)-f(x)<2$\\
(B) $\lim_{x\rightarrow\infty}f^{\prime}(x)=1$\\
(C) for all $x$ in the interval $[1,\infty),f(x+2)-f(x)>2$\\
(D) $f^{\prime}(x)$ is strictly decreasing in the interval $[1,\infty)$

Paper Questions

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