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The y-value f(c) is a local maximum value (also called relative maximum value) of f if there is an open interval containing the x-value c. When the graph of the function f, continuous at x=c, is increasing on the immediate left of the number x=c and decreasing on the immediate right of the number x=c, then the value of f at c is locally the largest, i.e., f(c) is a local maximum. This test can be extended to endpoints in the domain of f: if x=c is a left endpoint and decreasing on the immediate right, or if x=c is a right endpoint and increasing on the immediate left, then f(c) is a local maximum.

The y-value f(c) is a local minimum value (also called relative minimum value) of f if there is an open interval containing the x-value c. When the graph of the function f, continuous at x=c, is decreasing on the immediate left of the number x=c and increasing on the immediate right of the number x=c, then the value of f at c is locally the smallest, i.e., f(c) is a local minimum. This test can be extended to endpoints in the domain of f: if x=c is a left endpoint and increasing on the immediate right, or if x=c is a right endpoint and decreasing on the immediate left, then f(c) is a local minimum.

Any value of x in the domain of f is called a critical number of f' (also called critical point or critical value) if either f'(x)=0 or f'(x) does not exist. For continuous functions, the local maxima and local minima can only occur at the critical numbers or endpoints of the domain of f. These numbers separate the domain of f into intervals. At each critical number or endpoint, there are three possibilities.

- If the sign of f'(c) is positive on the left side (interval to the left) of a critical number of f' and negative on the right side, suggesting visually that the function is increasing on the left side and decreasing on the right side, then f has a local maximum at that critical number. If the sign of f'(c) is positive on the left side of a right endpoint or negative on the right side of a left endpoint, then f has a local maximum at that endpoint.
- If the sign of f'(c) is negative on the left side of a critical number of f' and positive on the right side, suggesting visually that the function is decreasing on the left side and increasing on the right side, then f has a local minimum at that critical number. If the sign of f'(c) is negative on the left side of a right endpoint or negative on the right side of a left endpoint, then f has a local minimum at that endpoint.
- If there is no change in sign of f'(c) from either side of a critical number to the other side, then the critical number is not a local maximum or local minimum for f. The curve has a horizontal tangent at the critical number but the point of tangency is not a turning point.

Read examples of optimization problems.