Does the description disclose side effects, auth requirements, rate limits, or destructive behavior?

Annotations already declare readOnlyHint=true and openWorldHint=false, so the safety profile is covered. The description adds one genuine behavioral caveat — classification is attempted only 'when possible', implying second-derivative test failure can occur — but it does not say what the tool returns in that inconclusive case. Useful but thin beyond the annotations.

Agents need to know what a tool does to the world before calling it. Descriptions should go beyond structured annotations to explain consequences.

Is the description appropriately sized, front-loaded, and free of redundancy?

One sentence, no filler, and the core operation is front-loaded before the classification clause. Nothing in it is redundant with the title or restates the name.

Shorter descriptions cost fewer tokens and are easier for agents to parse. Every sentence should earn its place.

Given the tool's complexity, does the description cover enough for an agent to succeed on first attempt?

An output schema exists, so return shape needn't be explained. However, for a calculus tool that depends on a second-derivative test with a stated 'when possible' caveat, the definition gives no hint about fallback behavior or the expected expression syntax, and offers no routing among overlapping calculus siblings.

Complex tools with many parameters or behaviors need more documentation. Simple tools need less. This dimension scales expectations accordingly.

Does the description clarify parameter syntax, constraints, interactions, or defaults beyond what the schema provides?

Schema description coverage is 100% with only two parameters, so the schema already documents 'expression' and 'variable' including the x default. The description's notation f' and f(x) reinforces that the expression is a single-variable function but adds no format, syntax, or variable-substitution detail beyond the schema. Baseline 3 applies.

Input schemas describe structure but not intent. Descriptions should explain non-obvious parameter relationships and valid value ranges.

Does the description clearly state what the tool does and how it differs from similar tools?

The description gives a specific verb ('Find') plus the mathematical resource (critical points where f'=0) and adds the classification behavior. An agent knows exactly what the tool computes, but nothing distinguishes it from the many calculus siblings such as calculator_derivative, calculator_second_derivative, or calculator_root_finder.

Agents choose between tools based on descriptions. A clear purpose with a specific verb and resource helps agents select the right tool.

Does the description explain when to use this tool, when not to, or what alternatives exist?

There is no when-to-use guidance and no named alternative. With over a hundred siblings, several of which overlap (derivative, second derivative, root finder, numerical solve), the agent is left to infer that this is the multi-step analysis tool rather than a single-step derivative or root solver.

Agents often have multiple tools that could apply. Explicit usage guidance like "use X instead of Y when Z" prevents misuse.