MATHEMATICS · Single correct · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
The function p(x)=(x−4)3 has a stationary point at x=4. Which statement best explains why this critical point is not a local maximum or minimum?
(a) The derivative changes from positive to negative at x=4.
(b) The derivative is zero at x=4, but its sign does not change because the factor (x−4)2 is always non-negative.
(c) The second derivative is negative at x=4, indicating that the point cannot be an extremum.
(d) Because p′′(4)=0, the second-derivative test fails, so the point cannot be a maximum or minimum.