Choices of $k$ for which $\operatorname{Var}(X-Y) \geq |\sigma_X-\sigma_Y|^k$ holds

Question

I have recently stumbled upon a question like this:

For which value of $k$ does the inequality $$\operatorname{Var}(X-Y) \geq |\sigma_X-\sigma_Y|^k$$ hold for all choices of random variables $X$ and $Y$ with finite variances ${\sigma^2_X}$ and ${\sigma^2_Y}$.

Options: $(A)\quad 1 \quad (B)\quad 2\quad (C)\quad 3\quad (D)\quad4$

I don't have any idea how to proceed. Any hint or help would be very much helpful.

Answer 1

Correlation coefficient $\frac{\text{Cov}(X,Y)}{\sigma_X\sigma_Y}$ of any random variables with finite variances does not exceed $1$, so
$$ \text{Cov}(X,Y) \leq \sigma_X\sigma_Y $$ Then use properties of variance to get $$ \text{Var}(X-Y) = \sigma_X^2 + \sigma_Y^2 - 2\text{Cov}(X,Y) \geq \sigma_X^2 + \sigma_Y^2 - 2\sigma_X\sigma_Y $$ This is sufficient to choose valid option from A,B,C,D.