How To Determine Relative Extrema Using The First Derivative Test In Calculus

By the 1st derivative test, a point is a relative minimum when

f’ changes from negative to positive

the derivative changes from negative to positive at that point.

To elaborate, let f(x) be a function whose derivative exists at a critical point c. To determine whether c is a relative minimum or maximum, we can use the 1st derivative test.

If the derivative of f changes sign from negative to positive at c, that is f'(c-) < 0 and f'(c+) > 0, then c is a relative minimum of the function. This means that the function decreases to the left of c and increases to the right of c, and c is the lowest point on the function in the local region around c.

Conversely, if the derivative changes from positive to negative at c, that is f'(c-) > 0 and f'(c+) < 0, then c is a relative maximum of the function. This means that the function increases to the left of c and decreases to the right of c, and c is the highest point on the function in the local region around c. If the derivative does not change sign at c, either because f'(c-) = f'(c+) = 0, or because the derivative is undefined at c, then c is not a local extremum of the function. It may still be a point of inflection or a vertical tangent, but further analysis is required to determine this.

More Answers: