If x y 1, which of the following must be true?

i. ln x ln y
ii. e^x e^y
iii. y^x x^y
iv. cos x cos y

A. (i...

Question

If x > y > 1, which of the following must be true?

i. ln x > ln y
ii. e^x > e^y
iii. y^x > x^y
iv. cos x > cos y

A. (i) and (ii)
B. (i) and (iii)
C. (iii) and (iv)
D. (ii) and (iv)

Accepted Answer

Statements (i) ln x > ln y and (ii) e^x > e^y must both be true; answer is A. When x > y > 1, we need to check four inequalities. The natural log and exponential functions are both strictly increasing for real numbers, so x > y implies ln x > ln y and e^x > e^y. The cosine function oscillates and is not monotone on (1, infinity), so cos x > cos y is not always true. The comparison y^x vs x^y depends on specific values and is also not always true.

If x > y > 1, which of the following must be true? i. ln x > ln y ii. e^x > e^y iii. y^x > x^y iv. cos x > cos y A. (i) and (ii) B. (i) and (iii) C. (iii) and (iv) D. (ii) and (iv)