Cutting insert, rotary tool, and method for manufacturing machined product

A cutting insert comprising:

• a main body extending from a first end toward a second end along a rotational axis, wherein
• the main body comprises
• a cutting edge positioned on a side of the first end,
• a rake face extending from the cutting edge toward the second end, and
• a flute extending from the rake face toward the second end,
• the rake face comprises
• a first surface region connected to the cutting edge and having a first rake angle,
• a second surface region positioned closer to the second end than the first surface region and having a second rake angle, and
• a third surface region positioned closer to the second end than the second surface region and having a third rake angle,
• the flute is positioned closer to the second end than the second surface region,
• the third surface region is adjacent to the flute rearward in a rotational direction about the rotational axis and on a side of an outer peripheral of the main body,
• the second rake angle is smaller than the first rake angle, and
• the third rake angle is smaller than the second rake angle.