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Tagged with cubics derivatives
36
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
1
vote
1
answer
60
views
Solving Bezier cubic derivative for t
Getting the derivative of the cubic Bezier curve:
$P(t)=P_0(1-t)^3+P_13t(1-t)^2+P_23t^2(1-t)+P_3t^3$
Produces the following:
$P'(t)=3(-P_0-2P_1)+6t(P_0+P_1+P_2)+3t^2(P_3-P_0)$
Assuming P'(t)=0, is ...
- 53
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
1
vote
0
answers
58
views
Meaning of $\left| (f\circ f)(x) \right|$ is not differentiable only on single point, when $f(x)$ is a cubic function?
$f(x)$ is a cubic function, and its leading coefficient is $1$.
$f(x)$ intersects with $x$-axis only on $(1,0)$.
$f'(1) \ne 0$.
$\left| (f\circ f)(x) \right|$ is not differentiable only on $x=2$.
...
- 343
2
votes
1
answer
69
views
Getting the area between $y$-axis, $f(x)$, $f(2-x)$ when both function is given by their subtraction?
For a polynomial $f(x)$, let function $g(x) = f(x) - f(2-x)$.
$g'(x) = 24x^2 - 48x + 50$
What is the area between $y=f(x)$, $y=f(2-x)$, and $y$-axis?
My approach:
$g(1) = f(1) - f(1) = 0$.
From $g'(...
- 343
2
votes
1
answer
49
views
Stuck on problems about differential / derivatives
For the real number $t$ that $t \ge 6 $, let $f(x) = \frac{1}{t} \left( \frac{1}{8}x^3 + \frac{t^2}{8}x + 2 \right)$.
Let the sum of all real numbers $k$ satisfying the following condition be $g(t)$ :
...
- 343
0
votes
0
answers
53
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Meaning of $ g(x) = f(x-1) \times \left|\lim_{ h\to 0} \frac{f(x+h)-f(x)}{h}\right| $ has only one non-differentiable point?
$f(x)$ is a cubic function with leading coefficient of 1,
$$ g(x) = f(x-1) \times \left|\lim_{ h\to 0}
\frac{f(x+h)-f(x)}{h}\right| $$
$g(x)$ is not differentiable only at $x= -1$.
$g(1)=0$, $f'(0)&...
- 343
1
vote
1
answer
69
views
What can be derived from the fact that $\lim_{x\to 0}\frac{f(x-3)f(x+3)}{x^3}$ is convergence?
Problem is :
There is a cubic function $f(x)$ , with its positive coefficient.
And,
$\lim_{x\to 0} {\frac{f(x-3)f(x+3)}{x^3}}$ is convergence.
there is only one natural number $k = k_1$ which makes ...
- 343
1
vote
3
answers
84
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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
0
votes
2
answers
45
views
Find a cubic function given the inflection point and minimal point
I came across this task that I just couldn't figure out.
The task gave me an extremum(1,1) and the inflection point(2,3), and I need to figure out the cubic function given the values.
Assuming the ...
- 25
0
votes
1
answer
181
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How do I find g(x) given the stationary points (2,9)?
question image
given the info I form the equation g(x) = AX³+BX²+AX+C and did g'(x) = 0 to find B = -13A/4 (when x = 2). So using B and putting it into g(x) = 9, I get AX³-13A/4X²+AX+C = 9 and I am ...
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
0
votes
3
answers
436
views
Find the derivative of the root of cubic equation
Consider the following cubic equation:
\begin{align}
\alpha ^3-2 \alpha ^2- \alpha(6+4x^2) +8 x^3-4x&=0
\end{align}
Where $x\in \mathbb{R}$.
The solutions $\alpha_1,\alpha_2,\alpha_3$ will depend ...
- 135
0
votes
2
answers
3k
views
Find a Cubic Function given an inflection point, critical point, and function value.
Find a cubic function f(x) = ax^3 + bx^2 + cx + d
Given:
Inflection point (0,18)
Critical point x = 2
F(2) = 2
I know how to solve for the general forms of the derivatives, and to set the values of ...
- 3
3
votes
1
answer
72
views
Proving $ 1+2f'(x)+\frac{2}{x(1+x^2)}\left(\frac{3x}{2}+f(x) \right)\ge \frac{6x^2}{1+8x^2} $.
Put
\begin{align*}
f(x)=\left( -\frac{x}{2} +\sqrt{\frac{1}{27}+\frac{x^2}{4}} \right)^{1/3}-\left( \frac{x}{2} +\sqrt{\frac{1}{27}+\frac{x^2}{4}} \right)^{1/3}
\end{align*}
Prove that
$$
g(x):=1+2f'...
- 2,950