Polar Curve and maximum width > Find the maximum width of the petal of the four-leaved rose $r = \cos2\theta$, which lies along the x-axis Here is the solution !enter image description here Can someone tell me how on earth did the solution come up with the first step? The statement " _The maximum width of the petal of the rose which lies on along the x-axis is twice the largest y-value of the curve on the interval..._ " Maybe the curve I sketched wasn't great, but I did it again on Mathematica and I still couldn't see how they notice this "ingenuous" subtle observation.

The petal lies along the positive $x$-axis. To say it differently, the positive $x$-axis is the axis of symmetry of the the petal. The _length_ of the petal is therefore measured horizontally, from the origin to the tip of the petal. The _width_ at any point is the size of the petal measured perpendicularly to the axis, i.e., at right angles to the $x$-axis. Thus, the width at any point is measured parallel to the $y$-axis and is therefore the distance between the top and bottom of the petal at that value of $x$. Since the petal is symmetric about the $x$-axis, that width is twice the $y$-value of the top edge of the petal. This reaches its maximum when the $y$-coordinate of the top edge of the petal is as large as possible.