Most often describes something with a sharp cutting edge or point, such as a knife, blade, or instrument. It is more formal and descriptive than everyday 鋭い.
I cut my finger with a sharp blade.
This task uses sharp tools.
Figurative use for a keen mind, incisive analysis, or acute perception; it tends to sound written or formal, and 鋭い is the more common everyday choice.
Her criticism is incisive and has no wasted words.
The perpetrator is believed to have used a sharp blade.
The wound showed a clean, sharp-cut edge.
With her sharp mind, she solved one difficult problem after another.
尖鋭 describes something sharply pointed or, figuratively, radical/extreme in stance, whereas 鋭利 stresses cutting keenness or mental incisiveness without the radical nuance.
切れ味 describes how well a blade actually cuts, while 鋭利 describes general sharpness or keenness, whether of an edge or a mind.
鈍い means dull or blunt, the direct opposite of 鋭利 whether describing a blade's edge or a person's mental sharpness.
Standard kanji spelling; mainly used in somewhat formal descriptions of sharp blades, tools, points, or keen perception.
A Sino-Japanese compound written with 鋭, associated with sharpness or keenness, and 利, associated here with sharpness or effectiveness. This is a component-based explanation of the written form, not a claim about the full historical derivation.
This reporter depicts the hidden side of society with keen observational insight.