Track / Stage 4 / 58 of 340 #318 of 2000

Problem 318

AMC 12 late, AIME early

Algebra Difficulty 4.6 Find the answer

* 4. The number of solutions to the equation $\sin ^{x} \alpha+\cos ^{x} \alpha=1\left(0<\alpha<\frac{\pi}{2}\right)$ is ( ).
(A) 0
(B) 1
(C) 2
(D) greater than 2

Your answer

Official solution

4. B Since $0<\sin \alpha, \cos \alpha<1$ , the function $f(x)$ $=\sin ^{2} \alpha+\cos ^{x} \alpha$ is decreasing, so the original equation will not have more than one root.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.