The curve $y(x) = ax ^ { 3 } + bx ^ { 2 } + cx + 5$ touches the $x$-axis at the point $P(-2,0)$ and cuts the $y$-axis at the point $Q$, where $y'$ is equal to 3. Then the local maximum value of $y(x)$ is
(1) $\frac { 27 } { 4 }$
(2) $\frac { 29 } { 4 }$
(3) $\frac { 37 } { 4 }$
(4) $\frac { 9 } { 2 }$

Let $f(x) = \int_0^x t(t-1)(t-2)\,dt$, $x > 0$. Then the number of points in the interval $(0, 3)$ at which $f(x)$ has a local maximum is $\_\_\_\_$.

Let $x = 2$ be a local minima of the function $f ( x ) = 2 x ^ { 4 } - 18 x ^ { 2 } + 8 x + 12 , x \in ( - 4,4 )$. If $M$ is local maximum value of the function $f$ in $( - 4,4 )$, then $M =$
(1) $12 \sqrt { 6 } - \frac { 33 } { 2 }$
(2) $12 \sqrt { 6 } - \frac { 31 } { 2 }$
(3) $18 \sqrt { 6 } - \frac { 33 } { 2 }$
(4) $18 \sqrt { 6 } - \frac { 31 } { 2 }$

The function $f ( x ) = 2 x + 3 x ^ { \frac { 2 } { 3 } } , x \in R$, has
(1) exactly one point of local minima and no point of local maxima
(2) exactly one point of local maxima and no point of local minima
(3) exactly one point of local maxima and exactly one point of local minima
(4) exactly two points of local maxima and exactly one point of local minima

Let $f ( x ) = 4 \cos ^ { 3 } x + 3 \sqrt { 3 } \cos ^ { 2 } x - 10$. The number of points of local maxima of $f$ in interval $( 0,2 \pi )$ is
(1) 3
(2) 4
(3) 1
(4) 2

The number of critical points of the function $f ( x ) = ( x - 2 ) ^ { 2 / 3 } ( 2 x + 1 )$ is
(1) 1
(2) 2
(3) 0
(4) 3

Let $f ( x ) = \int _ { 0 } ^ { x ^ { 2 } } \frac { \mathrm { t } ^ { 2 } - 8 \mathrm { t } + 15 } { \mathrm { e } ^ { t } } \mathrm { dt } , x \in \mathbf { R }$. Then the numbers of local maximum and local minimum points of $f$, respectively, are :
(1) 2 and 3
(2) 2 and 2
(3) 3 and 2
(4) 1 and 3

Let $a$ be a positive real number. We are to investigate local extrema of the function
$$f(x) = x^2 - 5 + 4a\log(2x + a + 8) \quad \left(-\frac{a}{2} - 4 < x < -2\right).$$
(1) When we differentiate the function $f(x)$ with respect to $x$, we obtain
$$f'(x) = \frac{\mathbf{A}(\mathbf{B}\, x + a)(x + \mathbf{C})}{\mathbf{D}\, x + a + \mathbf{E}}.$$
(2) Since a condition of $a$ is that $a > 0$ and the domain of $f(x)$ is $-\frac{a}{2} - 4 < x < -2$, the range of values of $a$ such that $f(x)$ has both a local maximum and a local minimum is
$$\mathbf{F} < a < \mathbf{G}.$$
In such a case, the sum of the local maximum and the local minimum is
$$\frac{a^2}{\mathbf{H}} + \mathbf{I} + \mathbf{IJ}\, a\log\mathbf{K}\, a.$$

Consider the real coefficient polynomial $f ( x ) = x ^ { 4 } - 4 x ^ { 3 } - 2 x ^ { 2 } + a x + b$. It is known that the equation $f ( x ) = 0$ has a complex root $1 + 2 i$ (where $i = \sqrt { - 1 }$). Select the correct options.
(1) $1 - 2i$ is also a root of $f ( x ) = 0$
(2) Both $a$ and $b$ are positive numbers
(3) $f ^ { \prime } ( 2.1 ) < 0$
(4) The function $y = f ( x )$ has a local minimum at $x = 1$
(5) The $x$-coordinates of all inflection points of the graph $y = f ( x )$ are greater than 0