Mastering The Second Derivative Test: How To Determine Relative Extremes In Math Functions

By the 2nd derivative test, a point is a relative minimum at x = c if

f'(c) = 0 and f” > 0

the second derivative evaluated at x = c is positive.

The second derivative test is a rule used to determine the nature of critical points (local extremes) of a function. Specifically, it helps us decide whether a critical point is a relative maximum or a relative minimum.

If f”(c) > 0, then the graph of the function is concave up (or smiling) at x = c. This means that the slope of the tangent line at x = c is increasing as we move to the right of x = c. So, if we choose any value of x slightly larger than c, the value of the function will increase relative to the value at x = c. Similarly, if we choose any value of x slightly smaller than c, the value of the function will decrease relative to the value at x = c. This indicates that x = c is a relative minimum of the function.

It’s important to note that this test only works for continuous functions whose second derivative is also continuous. If f”(c) = 0, the test is inconclusive and we may need to use other methods to determine the nature of the critical point.

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