Which of the following is (are) true of $f ( x ) = x ^ { 4 } - x ^ { 3 } ?$
(I) (0, 0) is an inflection point
(II) $\left( \frac { 1 } { 2 } , - \frac { 1 } { 16 } \right)$ is an inflection point
(III) (0, f(0)) is a relative minimum point
(IV) f is increasing on (0, 2)
A) I, II, and III
B) I, II, and IV
C) I and II
D) I, III, and IV
E) all of these

Question

Which of the following is (are) true of $f ( x ) = x ^ { 4 } - x ^ { 3 } ?$ 
(I) (0, 0) is an inflection point 
(II) $\left( \frac { 1 } { 2 } , - \frac { 1 } { 16 } \right)$ is an inflection point
(III) (0, f(0)) is a relative minimum point
(IV) f is increasing on (0, 2)
A) I, II, and III
B) I, II, and IV
C) I and II
D) I, III, and IV
E) all of these

Quizplus · Accepted Answer

To determine inflection points, we find the second derivative $f''(x)$ and check where it changes sign. For $f(x) = x^4 - x^3$, $f''(x) = 12x^2 - 6x$. Setting $f''(x) = 0$ gives $x = 0$ and $x = \frac{1}{2}$. Checking sign changes, both (0,0) and $\left(\frac{1}{2}, -\frac{1}{16}\right)$ are inflection points. For (III), $f'(x) = 4x^3 - 3x^2$, and $f'(0) = 0$, but $f''(0) = 0$ doesn't confirm a minimum. For (IV), $f'(x) > 0$ only for $x > \frac{3}{4}$, so f is not increasing on (0, 2).

Which of the Following Is (Are) True Of f(x)=x4−x3?f ( x ) = x ^ { 4 } - x ^ { 3 } ?f(x)=x4−x3?

Which of the Following Is (Are) True Of $f ( x ) = x ^ { 4 } - x ^ { 3 } ?$