How to Prove a Graph Has No Horizontal Tangent Lines

A horizontal tangent has slope zero. For the graph of a differentiable function $y=f(x)$, the tangent slope at $x=a$ is $f'(a)$. So the standard test for whether the graph has any horizontal tangents is to find out whether the derivative is ever zero.

The title alone does not specify the function, so it is not possible to give a proof about one particular graph. The method below applies once you have its equation.

The criterion

At an interior point $a$ of the function’s domain, the tangent line is

$$y=f(a)+f'(a)(x-a).$$

This line is horizontal exactly when its slope is zero:

$$f'(a)=0.$$

Thus, under the usual calculus definition, proving that the graph has no horizontal tangents means proving

$$f'(x)\ne 0$$

for every point in the relevant domain where the derivative exists. If the function is differentiable throughout the interior of its domain, this becomes a straightforward “derivative has no zeros” argument.

A step-by-step proof method

1. State the domain. Identify the points at which the function is defined and which points are under consideration. Horizontal tangents at endpoints may depend on whether one-sided tangents are included.
2. Differentiate. Calculate $f'(x)$, taking care to use the correct derivative rules.
3. Test for zeros. Set $f'(x)=0$ and solve, or show directly that the derivative cannot equal zero.
4. Use the domain restriction. If the derivative has possible zeros, check whether they belong to the domain. If none do, the graph has no horizontal tangent on that domain.
5. State the conclusion. Connect the result explicitly to the tangent slope: since $f'(x)\ne 0$ at every relevant point, no tangent there is horizontal.

Depending on the derivative, useful ways to rule out zeros include showing it is always positive or always negative, factoring it and checking each factor, or using an inequality to establish a strict sign.

Worked example

Consider $f(x)=x^3+x$, defined for all real $x$. Differentiate:

$$f'(x)=3x^2+1.$$

For every real $x$, $x^2\geq 0$, so $3x^2\geq 0$ and therefore

$$3x^2+1\geq 1>0.$$

The derivative is strictly positive everywhere, so it is never zero. Hence the graph of $f(x)=x^3+x$ has no horizontal tangent lines.

Notice that this proves more than the derivative is nonzero: it shows the function is increasing wherever it is defined. Monotonicity is not required for the method, though; the direct condition to verify is still that the derivative has no zero at any point in question.

Common mistakes

• Stopping after finding the derivative. A derivative formula by itself does not prove there are no horizontal tangents. You must establish that it cannot equal zero on the specified domain.
• Ignoring the domain. A derivative may have zeros outside the domain, or its formula may be undefined at some points. Check which points actually belong to the graph and what the problem asks you to consider.
• Confusing a horizontal tangent with a horizontal line elsewhere. The graph can intersect a horizontal line without having a horizontal tangent. Tangency is determined by the derivative at the point of contact.
• Claiming a derivative is positive without justification. For an expression such as $3x^2+1$, cite the facts $x^2\geq 0$ and $3x^2+1\geq 1$.
• Overlooking points where the derivative does not exist. In standard differential calculus, a horizontal tangent at $x=a$ is represented by a derivative of zero. If a problem uses a broader geometric definition of tangent, inspect nondifferentiable points separately and follow that definition.

If the derivative does have a zero

A zero of $f'$ is a candidate horizontal-tangent point, not automatically a maximum or minimum. For example, for $g(x)=x^3$, we have $g'(x)=3x^2$, so $g'(0)=0$. The tangent at the origin is horizontal even though the function does not have a local maximum or minimum there. The derivative test answers whether the tangent is horizontal; additional tests are needed to classify the point as an extremum.

To complete a proof for a particular graph, substitute its function into this process and analyze the derivative on the stated domain.