All Questions
67
questions
2
votes
1
answer
121
views
Three real roots of a cubic
Question: If the equation $z^3-mz^2+lz-k=0$ has three real roots, then necessary condition must be _______
$l=1$
$ l \neq 1$
$ m = 1$
$ m \neq 1$
I know there is a question here on stack about ...
- 497
2
votes
1
answer
76
views
Spivak, Ch, 25, 2(v): Is there some specific technique to factorize $x^3-x^2-x-2$ or must one guess that 2 is root?
The following problem appears in Ch. 25, "Complex Numbers" of Spivak's Calculus
2 (v) Solve the equation $x^3-x^2-x-2=0$.
Is there some specific technique to factorize this? Must one ...
- 5,021
2
votes
1
answer
135
views
Stationary points of a cubic function
If t is a positive constant, find the local maximum and minimum values of the function
$f(x) = (3x^2 - 4)\left(x - t + \frac{1}{t}\right)$
and show that the difference between them is $\frac{4}{9}(t + ...
- 33
3
votes
0
answers
92
views
About extreme values of $\{f(x)-x\}^2$ when $f(x)$ is a cubic function.
$t \ge 6$, $t \in \mathbb{R}$
$f(x) = \frac{1}{t}\left( \frac{1}{8}x^3 + \frac{t^2}{8}x+2\right)$
$\{f(x)-x\}^2$ has an extreme value on $x = k$
Sum of such $k = g(t)$
$g(p) = -1$ for some $p \in \...
- 343
6
votes
2
answers
248
views
Integration formula for cubic polynomial $\int_a^bq(x)dx=\frac{b-a}{2}(q(b)+q(a))-\frac{(b-a)^2}{12}(q'(b)-q'(a))$
Show that $\forall a,b\in \mathbb{R}$, with $a<b$, we have$$\int \limits _a^bq(x)\,dx=\frac{b-a}{2}(q(b)+q(a))-\frac{(b-a)^2}{12}(q'(b)-q'(a)),$$ where $q\in \mathcal{P}_3$ is a cubic polynomial.
I'...
- 545
1
vote
3
answers
84
views
Assume that $a=b$. If the function $f(x)$ is monotonously increasing, then (answer 1) $0<a<1$ (answer b).
The given function is $f(x)=x^3-3ax^2+3bx-2$.
I am aware that monotonously increasing means to continuously increase, so I tried getting this function's derivative and then setting it to zero, but to ...
- 149
15
votes
1
answer
354
views
Does the limit of the cubic formula approach the quadratic one as the cubic coefficient goes to $0$?
The formula for solving a cubic equation of the form $ax^3+bx^2+cx+d=0$ does not seem to yield the quadratic formula for the limit $\lim _{a \rightarrow 0} \text{(cubic formula)}$.
But, if one tries ...
- 297
3
votes
2
answers
80
views
Minimum value of M such that the cubic modulus value is always less than M for x in between -1 and 1 both included
Minimum value of $M$ such that $\exists a, b, c \in \mathbb{R}$ and
$$
\left|4 x^{3}+a x^{2}+b x+c\right| \leq M ,\quad \forall|x| \leq 1
$$
What i considered was that putting x= 0, 1 and -1 we get ...
- 431
0
votes
0
answers
116
views
Finding the roots of an absolute value natural log function, regarding the usage of Newton-Raphson
This is an exercise I saw in my last test, I've been practicing it but I'm currently a bit lost when finding the roots, here's the exercise:
Given $f:f(x)=-x^2+2x+3$, be F/F(x) is a primitive of f ...
- 31
2
votes
1
answer
304
views
What do the inflection points of complex functions look like when plotted?
The inflection points of real continuous functions have a relatively clear visual interpretation - they are the points when the function's graph moves from convex to concave, or vice-versa. Conversely,...
- 370
0
votes
1
answer
76
views
Applying general cubic formula to example
I'm trying to understand the general cubic formula so let's try applying it to an example:
Let's take:
$$(x-1)(x-2)(x-3) = 0$$
$$\implies x^3 - 6x^2 +11x -6 = 0$$
So the general cubic formula should ...
- 2,660
0
votes
0
answers
63
views
Understanding the general cubic formula? ("Changing the choice of a cubic root"?)
Paraphrasing Wikipedia > Cubic Equation > General cubic formula, and ignoring special cases and caveats, it says:
The cubic equation:
$$ax^3 + bx^2 + cx + d = 0$$
can be solved as follows:
Let:
$...
- 2,660
1
vote
1
answer
769
views
The roots of a cubic auxiliary equation are $𝑚_1 = 4$ and $𝑚_2 = 𝑚_3 = 5$. What is the corresponding homogeneous linear differential equation?
Is your solution unique?
My try:
$m-4=0, m-5=0 \text { and } m-5=0$
$(m-4)(m-5)^{2}=0$
$(m-4)(m^{2}-10m+25)=0$
$m^{3}-10m^{2}+25m-4m^{2}+40m-100=0$
$m^{3}-14m^{2}+65m-100=0$
$d^{3}y/dx^{3}-14d^{2}y/dx^...
-1
votes
2
answers
198
views
Cubic's derivative
For a cubic $ax^3+bx^2+cx+d$, the radicand of the derivative is $b^2-3ac$ and I was wondering if there's a word for this term. I know that the nature of it being positive, zero, or negative will ...
user612996
1
vote
1
answer
79
views
Is there a simple reason for why the two stationary points in a cubic polynomial has its second derivative equal in magnitude but opposite in sign.
Given a polynomial $f(x) = x^3 + ax^2 + bx + c$, then
$$f'(x) = 3x^2 + 2ax + b$$
$$f''(x) = 6x + 2a$$
The stationary points in the curve will have $x = \frac{-a \pm \sqrt{a^2 - 3b}}{6}$, but then the ...
- 1,447