How many real solutions are there to the equation 3cos x = √x where x is in radians?
- A0
- B1
- C2
- D3
- E4
- F5
- Ginfinitely many
Show the answer and worked solution
answer · D
- A0
- B1
- C2
- D3
- E4
- F5
- Ginfinitely many
Sketch both sides. √ x needs x ≥ 0 and rises steadily, while 3cos x oscillates between −3 and 3; once √ x > 3, that is x > 9, they can never meet again, so all solutions lie in [0, 9]. On [0, π2] the cosine curve falls from 3 to 0 while √ x rises from 0, so they cross once. On [π2, 3π2] the cosine is negative and √ x positive, so no crossings. On [3π2, 5π2] the cosine climbs to 3 at x = 2π, which beats √2π ≈ 2.51, then falls back to 0 — one crossing on the way up and one on the way down. Beyond 5π2 ≈ 7.85 the cosine is negative again, so there are 3 solutions in all.