Monotonicity and Sign of Derivative

If the sign of the derivative of a real function is unchanging, the function is monotonic, either increasing or decreasing depending on the sign. This should be intuitive from the way the derivative is defined, since a positive derivative at all points on some interval corresponds with an increasing tangent at all such points.

Positive Derivative Corresponds with Strictly Increasing Function

If f is continuous on [a,b] and f′(x)>0∀x∈(a,b), then for all y,z∈(a,b):

y<z⟹f(y)<f(z).

That is f is monotonic increasing.

Proof

Because differentiability implies continuity, given that y,z∈(a,b) where f is differentiable, f is continuous on [a,b].

Applying the mean value theorem to f over [y,z], there exists a c∈(a,b) such that for all y,z∈(a,b):

f(y)−f(z)y−z=f′(c)>0.

Given that y<z, y−z<0 and therefore we must reverse the inequality through multiplication, yielding

f(y)−f(z)<0⟹f(y)<f(z).
Negative Derivative Corresponds with Strictly Decreasing Function

If f is continuous on [a,b] and f′(x)<0∀x∈(a,b), then for all y,z∈(a,b):

y<z⟹f(y)>f(z).

That is f is monotonic decreasing.

Non-Negative Derivative Corresponds with Non-Decreasing Function

If f is continuous on [a,b] and f′(x)≥0∀x∈(a,b), then for all y,z∈(a,b):

y<z⟹f(y)≤f(z).

That is f is monotonic non-increasing.

Non-Positive Derivative Corresponds with Non-Increasing Function

If f is continuous on [a,b] and f′(x)≤0∀x∈(a,b), then for all y,z∈(a,b):

y<z⟹f(y)≥f(z).

That is f is monotonic non-increasing.