How To Find Critical Points Of A Function: A Step-By-Step Guide

find points where tangent line is horizontal AKA

where derivative is 0, answer should be two points

Points at which the tangent line is horizontal are called critical points. To find them, you need to find the derivative of the function and set it equal to zero.

1. Find the derivative of the function.
If the function is f(x), then the derivative is f'(x).

2. Set the derivative equal to zero.
f'(x) = 0

3. Solve for x.
You can do this algebraically or graphically. If algebraically, you can use techniques such as factoring, the quadratic formula or solving equations with exponents. If graphically, you can plot the function and look for the x-values where the curve crosses the x-axis, or where there are cusps, maxima or minima.

4. Check for extraneous solutions.
Some equations may have solutions that do not make sense in the context of the problem. For example, if the original function is a distance or time function, negative values for x may not make sense. Make sure to check your solutions against the original problem to eliminate any that do not make sense.

5. Find the y-coordinates of the critical points.
Plug the x-values found in step 3 into the original function to get the corresponding y-values. These are the coordinates of the critical points.

6. Interpret the results.
If the function is a graph of a physical phenomenon or real-life situation, the critical points may represent important turning points or inflection points. If the function is a mathematical expression, the critical points may represent points of interest in the graph, such as maxima or minima.

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