For 1 3 < x < y < 2, which of the following statements is/are always correct? 1. x + 1 x < y + 1 y 2. √1+y^2 y < √1+x^2 x Select the answer using the…

The correct option is B - II only.

Solution

We are given the range: $1/3 < x < y < 2$1/3<x<y<2. We need to check if the statements are always correct.

Evaluating Statement I: $x + 1/x < y + 1/y$x+1/x<y+1/y
Let $f(t) = t + 1/t$f(t)=t+1/t. To see if $f(x) < f(y)$f(x)<f(y) when $x < y$x<y, we check if the function is increasing.
The derivative $f'(t) = 1 - 1/t^{2}$f′(t)=1−1/t2.
- If $t > 1$t>1, $f'(t) > 0$f′(t)>0 (Function is increasing).
- If $t < 1$t<1, $f'(t) < 0$f′(t)<0 (Function is decreasing).
Since our range (1/3 to 2) includes values both below and above 1, the inequality will not always hold.
Example: Let $x = 1/2$x=1/2 and $y = 1$y=1.
$x + 1/x = 0.5 + 2 = 2.5$x+1/x=0.5+2=2.5
$y + 1/y = 1 + 1 = 2$y+1/y=1+1=2
Here, $2.5 < 2$2.5<2 is False. Thus, Statement I is not always correct.

Evaluating Statement II: $\sqrt{1+y^{2}}/y < \sqrt{1+x^{2}}/x$1+y2​/y<1+x2​/x
Let $g(t) = \sqrt{1+t^{2}}/t$g(t)=1+t2​/t. We can rewrite this as $g(t) = \sqrt{(1+t^{2})/t^{2}} = \sqrt{1/t^{2} + 1}$g(t)=(1+t2)/t2​=1/t2+1​.
As t increases:
1. t^2 increases.
2. $1/t^{2}$1/t2 decreases.
3. $(1/t^{2} + 1)$(1/t2+1) decreases.
4. $\sqrt{1/t^{2} + 1}$1/t2+1​ decreases.
Since g(t) is a strictly decreasing function for all $t > 0$t>0, if $x < y$x<y, then $g(x) > g(y)$g(x)>g(y).
This matches the statement: $g(y) < g(x)$g(y)<g(x).
Therefore, Statement II is always correct.

Why the other options are incorrect

• Option (a) - I only: This is incorrect because Statement I fails when x and y are chosen such that x is significantly less than 1 (where the function is decreasing) and y is closer to 1.
• Option (c) - Both I and II: This is incorrect because, as demonstrated above, Statement I is not universally true for the given interval.
• Option (d) - Neither I nor II: This is incorrect because Statement II is mathematically certain due to the strictly decreasing nature of the function $\sqrt{1/t^{2} + 1}$1/t2+1​ for positive values.

Key Concept

Monotonicity of functions: A function $f(x)$f(x) is increasing if its derivative is positive, and decreasing if its derivative is negative, determining the direction of inequalities.