Question

For the function $f(x)=xcos$ $\frac{1}{x},xâ‰\yen 1$,

• A. for at least one $x$ in the interval
  $[1,∞),f(x+2)−f(x)<2$
• B. $li{m}_{x→∞}{f}^{′}(x)=1$
• C. for all $x$ in the interval $[1,∞),f(x+2)−f(x)>2$
• D. $f^{′}(x)$ is strictly decreasing in the interval $[1,∞)$

Answer 1

Answer: (B), (C), (D)

Given, f (x) = x cos $\frac{1}{x},xâ‰\yen 1⇒f{}^{′}(x)=\frac{1}{x}sin\frac{1}{x}+cos\frac{1}{x}$
$⇒$ f " (x) = - $\frac{1}{x^3}cos\left(\frac{1}{x}\right)$
Now, $li{m}_{x→∞}$ f ' (x) = 0 + 1 = 1 $⇒$ Option (b) is correct
Now, $x∈[1,∞)⇒\frac{1}{x}∈(0,1]⇒$ f " (x) < 0
Option (d) is correct.
As f ' (1) = sin 1 + cos 1 > 1
f ' (x) is strictly decreasing and $li{m}_{x→∞}$ f ' (x) = 1
So, graph of f ' (x) is shown as below
Now, in
[ x, x + 2], x $∈[1,∞),$ f (x) is
continuous and differentiable
so by LMVT,
f ' (x) = $\frac{f(x+2)−f(x)}{2}$
As, f' (x) > 1
For all x $∈[1,∞)$
$⇒\frac{f(x+2)−f(x)}{2}>1⇒$ f (x + 2) - f (x) > 2
For all x $∈[1,∞)$