Horizontal Tangent Line Calculator

Enter a function f(x) and the Horizontal Tangent Line Calculator finds all points where the tangent line is horizontal — where the derivative equals zero. Input your function expression (e.g. x^2 + 3x, 2x^3 + 3x^2 - 12x + 1) and get back the critical x-values, the corresponding y-values, and the horizontal tangent line equations (y = c) at those points.

Enter a polynomial function using x as the variable. Use ^ for exponents and * for multiplication. Examples: x^2 + 5, 2*x^3 + 3*x^2 - 12*x + 1

Lower bound of the x range to search for horizontal tangent points

Upper bound of the x range to search for horizontal tangent points

Results

Horizontal Tangent Points Found

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Critical x-Values (f'(x) = 0)

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Horizontal Tangent Line Equations

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Derivative f'(x)

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Function and Horizontal Tangent Points

Results Table

Frequently Asked Questions

What is a horizontal tangent line?

A horizontal tangent line is a line tangent to a curve at a point where the slope is zero. This occurs when the derivative of the function f'(x) = 0. Geometrically, the curve momentarily goes neither up nor down at that point — it is perfectly flat.

How do I find horizontal tangent lines of a function?

To find horizontal tangent lines, take the derivative f'(x) of the function, set it equal to zero, and solve for x. Substitute each x-value back into the original function f(x) to find the corresponding y-value. The horizontal tangent line at each point is simply y = f(x) at that x.

What types of functions can I enter into this calculator?

This calculator supports polynomial functions of any degree entered as expressions in x. Use standard notation: ^ for exponents (x^2), * for multiplication (3*x), and + / - for addition and subtraction. Examples include x^2 + 5, 2*x^3 + 3*x^2 - 12*x + 1, and x^4 - 4*x^2.

Are all points where f'(x) = 0 local maxima or minima?

Not necessarily. Points where f'(x) = 0 are called critical points, and they can be local maxima, local minima, or inflection points (saddle points). To classify them, apply the second derivative test: if f''(x) > 0 the point is a local minimum, if f''(x) < 0 it is a local maximum, and if f''(x) = 0 the test is inconclusive.

Why does a cubic function like 2x³ + 3x² - 12x + 1 have two horizontal tangent points?

A cubic polynomial has a degree-3 derivative that is degree-2 (a quadratic). A quadratic equation can have up to two real roots, which is why cubic functions can have up to two points where f'(x) = 0 — one local maximum and one local minimum — each with a horizontal tangent line.

What does the equation of a horizontal tangent line look like?

A horizontal tangent line always takes the form y = c, where c is a constant equal to the y-value of the function at the critical point. For example, if f(2) = 7 and f'(2) = 0, then the horizontal tangent line at x = 2 is y = 7.

Can a function have no horizontal tangent lines?

Yes. If the derivative f'(x) has no real roots — meaning the equation f'(x) = 0 has no real solutions — then the function has no horizontal tangent lines within the real numbers. For example, f(x) = e^x has a derivative of e^x which is always positive and never zero.

How does the search range affect the results?

The search range limits the x-interval over which the calculator looks for critical points. Horizontal tangent points outside the specified range will not be shown. If you suspect tangent points exist outside the default range of -10 to 10, expand the search range accordingly.

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