All Questions
Tagged with cubics inequality
26
questions
5
votes
3
answers
160
views
Prove $2(a+b+c)\left(1+\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)\ge 3(a+b)(b+c)(c+a)$ for $abc=1.$
Let $a,b,c>0: abc=1.$ Prove that: $$2(a+b+c)\left(1+\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)\ge 3(a+b)(b+c)(c+a). $$
I've tried to use a well-known lemma but the rest is quite complicated for me.
...
3
votes
1
answer
163
views
Prove $(1+a^3) (1+b^3)(1+c^3) \ge (\frac{ab+bc+ca+1}{2})^3$
This is a question from 4U maths (highest level of Y12 maths in Australia) from a generally difficult paper. The question itself does not define what a, b, and c are - based on the comments, we assume ...
- 33
1
vote
1
answer
76
views
Best way to solve $\frac{x^3+3}{x^2+1}>\frac{x^3-3}{x^2-1}$
I was wondering what the best way to solve questions like these are?
$$\frac{x^3+3}{x^2+1}>\frac{x^3-3}{x^2-1}$$
I can get the answer, which is $(-\infty,-1)\cup(1,3)$. But I'm not sure if I have ...
- 21
1
vote
2
answers
131
views
How do I solve $x^3 - 2x + 2 \ge 3 - x^2$?
I believe there will be values of $x$ for which the inequality $x^3 - 2x + 2 \ge 3 - x^2$ is true and values for which it is not true, because:
LHS asymptotically increases but RHS decreases for ...
- 139
1
vote
2
answers
81
views
How to determine the order of the real roots of a cubic equation?
This is a self-answered question (I didn't find a reference, and thought of documenting this). Consider the equation
$$
t^3+pt+q=0.
$$
Its discriminant is
$$
\Delta=-(4p^3+27q^2).
$$
Suppose that it ...
- 25.3k
1
vote
2
answers
89
views
Is this quadratic polynomial monotone at solutions of this cubic?
Let $0<s < \frac{4}{27}$. The equation $x(1-x)^{2}=s$ admits exactly two solutions in $(0,1)$: Denote by $a,b$ be these solutions, and suppose that $a<b$.
Does
$$
(1-a)^2+2a^2<(1-b)^2+2b^2
...
- 25.3k
-3
votes
1
answer
48
views
How is $\xi+2a\eta<0$ an "obvious necessary condition" for $y^3+2y^2(1-2a-\xi)+y(1-4\xi+8a\xi)-2\xi-4a\eta >0$ to be satisfied for positive $y$?
How is $$\xi+2\alpha\eta<0$$ an 'obvious necessary condition' for the inequality
$$y^3+2y^2(1-2\alpha-\xi)+y(1-4\xi+8\alpha\xi)-2\xi-4\alpha\eta >0$$
to be satisfied for positive $y$ (as claimed ...
1
vote
2
answers
94
views
Is the theory of $\Delta\le0$, true in cubic functions$?$
If $ax^2+\frac{b}{x}\ge c$ $\forall x>0$ where $a>0 \:\: , b>0$ Show that $27ab^2\ge4c^3$
My work:
Let a function $f(x)$ be $$ax^2+\frac{b}{x}\ge c$$ or we can rewrite $f(x)$ as $$ax^3-cx+b\...
user1080568
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
4
answers
206
views
Unreal root of quadratic equation
Set a positive real number such that
$$a^3=6(a+1)$$
Prove that the equation
$$x^2 + ax + a^2 -6 = 0$$ there is no real solution
Solution attempt:
condition : $$a^2 - 4a^2 + 24 <0$$
$$a^2>8$$
$$a&...
- 77
0
votes
2
answers
147
views
Prove that $2a + 2ab + abc \leq 18$ when $a + b + c = 5$ where $a, b, c \in \mathbb{R}$ without calculus.
I have already had help with proving the conditional statement on this post.
Apparently, it includes the following trick from this website, and it says it works for any cubic polynomial function.
...
- 115
3
votes
1
answer
72
views
Proving $ 1+2f'(x)+\frac{2}{x(1+x^2)}\left(\frac{3x}{2}+f(x) \right)\ge \frac{6x^2}{1+8x^2} $.
Put
\begin{align*}
f(x)=\left( -\frac{x}{2} +\sqrt{\frac{1}{27}+\frac{x^2}{4}} \right)^{1/3}-\left( \frac{x}{2} +\sqrt{\frac{1}{27}+\frac{x^2}{4}} \right)^{1/3}
\end{align*}
Prove that
$$
g(x):=1+2f'...
- 2,950
0
votes
0
answers
48
views
Is there a generic way to solve cubic inequalities?
Hi Pardon me if it has already been asked. I could not find any material on math-stack, hence asked this question!
I have a cubic inequality of the form,
\begin{equation}
-a_1x^3+a_2x^2+a_3x+a_4<...
- 153
2
votes
2
answers
109
views
Find the minimum value of $a^2+b^2+c^2+2abc$ when $a+b+c=3$ and $a,b,c\geq0$.
Given $a,b,c\geq0$ such that $a+b+c=3$, find the minimum value of $$P=a^2+b^2+c^2+2abc.$$
It seems like the minimum value of $P$ is $5$ when $a=b=c=1$, but I can find at least one example where $P<...
- 235
3
votes
2
answers
197
views
$x^{3}+ax^2+bx+c$ has all roots negative real numbers and a<3. Establish an inequality between only b and c [duplicate]
A cubic equation $x^{3}+ax^2+bx+c$ has all negative real roots and $a, b, c\in R$ with $a<3.$
Prove that $b+c<4.$
My attempt :
Let the cubic be $f(x)$
Plotting graph we see that ,
$f(x\geq 0)&...
- 171