The second derivative test reveals that $f ( x ) = x ^ { 2 } - 3$ has
A) a relative maximum at $x = \sqrt { 3 }$
B) a point of inflection at $x = \sqrt { 3 }$
C) a relative minimum at x = 0.
D) a point of inflection at x = 3.

Question

Quizplus · Accepted Answer

The second derivative of $f(x) = x^2 - 3$ is $f''(x) = 2$, which is positive for all $x$, indicating concavity upwards everywhere and no points of inflection. Since the first derivative $f'(x) = 2x$ changes sign from negative to positive at $x = 0$, this point is a relative minimum, not a maximum or a point of inflection.

The Second Derivative Test Reveals That f(x)=x2−3f ( x ) = x ^ { 2 } - 3f(x)=x2−3

The Second Derivative Test Reveals That $f ( x ) = x ^ { 2 } - 3$