examday

/TMUA

Consider the following attempt to solve an equation. The steps have been numbered for reference.

Equation: $3\cos x=\sqrt{x}$, where $x$ is in radians.

Solving Steps:

• (1) Start with the equation: $3\cos x=\sqrt{x}$.
• (2) Square both sides of the equation: $9\cos^2x=x$.
• (3) Use the identity $\cos^2x=\dfrac{1}{2}(1+\cos2x)$ to rewrite the equation as $9\cdot\dfrac{1}{2}(1+\cos2x)=x$.
• (4) Graph the functions $y=\dfrac{1}{2}(1+\cos2x)$ and $y=\dfrac{x}{9}$.
• (5) From the graph, observe that at $x=\pi$, $y=\pi/9\approx0.35$ is less than $y=\dfrac{1}{2}(1+\cos2\pi)=1$, and at $x=2\pi$, $y=2\pi/9\approx0.70$ is less than 1. The graphs intersect at five points.
• (6) Conclude that the equation has five solutions.

Which one of the following statements is true?