First slide

Maxima and minima

Question

The set of value(s) of a for which the function $f (x) = \frac{{ax}^{3}}{3} + (a + 2) x^{2} + (a - 1) x + 2$ possesses a negative point of inflection is

Moderate

A

$(- \infty, - 2) \cup (0, \infty)$
B

${- 4 / 5}$
C

$(- 2, 0)$
D

empty set

Solution

$\begin{array}{r} f^{'} (x) = {ax}^{2} + 2 (a + 2) x + (a - 1) \\ f^{''} (x) = 2 ax + 2 (a + 2) = 0 \end{array}$
Thus, $x = - \frac{a + 2}{a}$ which is the point of inflection
Given that we must have $\frac{a + 2}{a} < 0 or a \in (- \infty, - 2) \cup (0, \infty)$ .

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