The second derivative test reveals that f (x) = x² -8 has A) nothing significant at x = 8. B) a relative minimum at x = 0. C) a relative maximum at $x = \sqrt { 8 }$ . D) a point of inflection at $x = \sqrt { 8 }$ .

Question

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The second derivative test involves finding the first and second derivatives of the function, then analyzing the sign of the second derivative at critical points (where the first derivative is zero or undefined) to determine concavity and thus identify relative maxima or minima. For $f(x) = x^{12} - 8$, the first derivative $f'(x) = 12x^{11}$ is zero at $x = 0$, indicating a critical point. The second derivative $f''(x) = 132x^{10}$ is positive at $x = 0$, indicating that the function is concave up at this point, thus confirming a relative minimum at $x = 0$. Choices A, C, and D do not correctly interpret the results of applying the second derivative test to this function.

The Second Derivative Test Reveals That F (X) = X2 x=8x = \sqrt { 8 }x=8​

The Second Derivative Test Reveals That F (X) = X² $x = \sqrt { 8 }$