<s>[INST] What is the relative maximum or minimum and the point of inflection? 
$f(x) = ax^3+bx^2+cx +d$, determine a, b, c, and d such that the graph of $f$ has a extreme in $(0,3)$ and a point of inflection in $(-1,1)$.

When is a quadratic I know that the formula $V=(\frac{-b}{2a},\frac{-\triangle}{4a})$ gives the maximum or minimum of the function, but for $f(x) = ax^3+bx^2+cx +d$, not seems the same. I tried to make $f(x) = x(ax^2+bx+c) +d$ but doesn't work and for the point of inflection  I know that is the point of the second derivative is equal zero and change of the signal, but how I can make the function change in (-1,1)?
 [/INST] Extremum at $(0,3)$:
We need $f'(0) = 0$ and $f''(0) \ne 0$ for a local minimum or maximum at $x=0$, so
$$
f'(x) = 3ax^2 + 2bx + c \Rightarrow
f'(0) = c = 0 \\
f''(x) = 6ax + 2b \Rightarrow
f''(0) = 2b \ne 0
$$
then $f(0) = 3$ was required:
$$
f(0) = d = 3 
$$
Inflection point at $(-1, 1)$:
Further we need $f''(-1) = 0$, because the curvature
$$
k(x)= \frac{f''(x)}{(1+f'(x)^2)^{3/2}}
$$ 
is required to change sign for a point of inflection, and it has the same sign as $f''$, so
$$
f''(x) = 6ax + 2b \Rightarrow
f''(-1) = -6a + 2b = 0
$$
and $f(-1) = 1$ was required:
$$
f(-1) = -a + b -c + d = -a + b+3 = 1
$$
so we need to solve
$$
-6a + 2b = 0 \\
-a + b + 3 = 1
$$
which is equivalent to
$$
b = 3a \\
-a + 3a = -2 
$$
which is equivalent to
$$
a = -1 \\
b = -3
$$
Checking the solution:
This gives
$$
(a, b, c, d) = (-1, -3, 0, 3)
$$
which is the function
$$
f(x) = -x^3 - 3x^2 + 3
$$
It has the derivatives
$$
f'(x) = -3x^2 -6x \\
f''(x) = -6x - 6
$$
with $f'(0) = 0$, $f''(0) = -6 \ne 0$, $f(0) = 3$, so we have a local maximum at $(0,3)$.
Further it has $f''(-1) = 0$, $f(-1) = 1$. Then we have for $\epsilon > 0$
$$
f''(-1 - \epsilon) = -6(-1-\epsilon) - 6 = 6 \epsilon > 0 \\
f''(-1 + \epsilon) = -6(-1+\epsilon) - 6 = -6 \epsilon < 0
$$
so we have a change of sign of $f''$ and thus also for the curvature $k$ around $x=-1$, which makes $(-1,1)$ a point of inflection.

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