Critical Point

Definition

A critical point is a point where the derivative is zero or undefined.

For multi-input functions, it is where the gradient is zero or undefined.

Basic idea:

local change becomes flat or unclear

A critical point is a candidate location for special behavior.

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Derivative Equals Zero

If:

f'(x) = 0

then the function is locally flat at that point.

Example shape:

decreasing -> flat -> increasing

This may indicate a local minimum.

But flatness alone does not guarantee a minimum.

Zero derivative means local flatness, not automatic success.

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Gradient Equals Zero

For a multi-input function:

gradient = [0, 0]

means the function has no first-order increase in the coordinate directions at that point.

The surface may be:

bottom of a bowl
top of a hill
saddle-like
flat plateau

Zero gradient identifies a candidate, not the type of point.

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Undefined Derivative

A point can also be critical if the derivative is undefined.

This can happen at:

sharp corners
cusps
breaks

Example idea:

function has a corner at x = 0

The slope from the left and right may not match.

Some important points are not smooth.

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Candidate, Not Conclusion

A critical point needs further interpretation.

It could be:

local minimum
local maximum
saddle point
flat region

So the question is not only:

Is the derivative zero?

It is also:

What kind of behavior exists around this point?

Critical point means "look here more carefully."

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Why This Matters

When improving a function, critical points are important because local movement may slow down or stop.

If the gradient is near zero, the local signal is weak.

That can mean:

near a useful minimum
stuck near a poor flat region
passing through a saddle

Flatness must be interpreted by surrounding behavior.

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Final Summary

• A critical point occurs where the derivative or gradient is zero or undefined.
• Zero derivative means local flatness.
• Undefined derivative can occur at sharp or broken points.
• A critical point may be a minimum, maximum, saddle, or plateau.
• It is a candidate for special behavior, not a final diagnosis.