Checking if an interval value satisfies an expression

In summary, the conversation discusses a quartic polynomial with real roots that has a specific interval of possible roots. The question is whether at least one of these roots satisfies a given inequality. The conversation suggests using the Samuelson-Laguerre inequality to find the interval for the roots, but this results in strong constraints on one of the variables. The speaker is looking for a more efficient method to prove the inequality.
  • #1
Siron
150
0
Hello!

Suppose a quartic polynomial with real roots where all its roots lie in the interval
$$x \in \left[\frac{5}{8}a-\frac{3}{8}\sqrt{15a^2-16b}, \frac{5}{8}a+\frac{3}{8}\sqrt{15a^2-16b}\right]$$
where $a \in \mathbb{R}$ and $b<(15/16)a^2$. Is there a way to check that at least one of those roots satisfies the inequality:
$$\frac{5ax-4x^2-b}{15ax-20x^2-3b}<0.$$

I solved the above inequality with wolfram alpha which displays the following solutions (after summarizing cases and taking $b<(15/16)a^2$ into account):
$$x \in \left[\frac{5a}{8}-\frac{1}{8}\sqrt{25a^2-16b}, \frac{3a}{8}-\frac{1}{8}\sqrt{\frac{3}{5}}\sqrt{15a^2-16b}\right]$$
or
$$x \in \left[\frac{3a}{8}+\frac{1}{8}\sqrt{\frac{3}{5}}\sqrt{15a^2-16b}, \frac{5a}{8}+\frac{1}{8}\sqrt{25a^2-16b} \right].$$

So ... in fact I think I only need to check wether the interval for my roots lies in one of the above intervals. However, I think this will result in too strong restrains on $b$. Is there a better alternative to solve the question?

Many thanks!
Kind regards,
Siron
 
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  • #2
I now see that my question is not that clear so I will give more details. The quartic polynomial which I refer to in my previous post is given by:
$$f(x) = 16x^4-40ax^3+(15a^2+24b)x^2-18abx+3b^2$$
where $a,b \in \mathbb{R}$. If the discriminant is positive and $b<(15/16)a^2$ then the quartic has four real roots (it's not difficult to derive that). So I'm wondering if its possible to prove that at least one of these real roots satisfies the inequality:
$$\frac{5ax-4x^2-b}{15ax-20x^2-3b}<0.$$
or equivalently lies in one of the two intervals:
$$x \in \left[\frac{5a}{8}-\frac{1}{8}\sqrt{25a^2-16b}, \frac{3a}{8}-\frac{1}{8}\sqrt{\frac{3}{5}}\sqrt{15a^2-16b}\right] := [l_1,r_1] $$
or
$$x \in \left[\frac{3a}{8}+\frac{1}{8}\sqrt{\frac{3}{5}}\sqrt{15a^2-16b}, \frac{5a}{8}+\frac{1}{8}\sqrt{25a^2-16b} \right] := [l_2,r_2].$$

I first thought about the intermediate value theorem and was thinking if I maybe could force $f(l_1)<0$ and $f(r_1)>0$ (or $f(l_2)<0$ and $f(r_2)>0$). However this gives very complicated expressions. Next, I tried to use the Samuelson-Laguerre inequality. Basically, this inequality implies an interval for the roots of the quartic. Applying the inequality, the four real roots of the quartic have to lie in the interval:
$$x \in \left[\frac{5}{8}a-\frac{3}{8}\sqrt{15a^2-16b}, \frac{5}{8}a+\frac{3}{8}\sqrt{15a^2-16b}\right]$$
and this was in fact my question/idea of the first post. Maybe I can force conditions such that the above interval lies within $[l_1,r_1]$ or $[l_2,r_2]$ but this gives very strong constraints on $b$.

Therefore, I'm wondering if there is another more efficient idea/method ... ?

Thanks!
 

FAQ: Checking if an interval value satisfies an expression

What does it mean to "check if an interval value satisfies an expression"?

Checking if an interval value satisfies an expression means determining whether a given value falls within a specified range or interval and also satisfies a given mathematical expression or condition.

How do I check if an interval value satisfies an expression?

To check if an interval value satisfies an expression, you can plug in the value into the expression and see if it results in a true or false statement. Alternatively, you can graph the expression and the interval value on a number line to visually see if the value falls within the interval and also satisfies the expression.

What are some common examples of checking if an interval value satisfies an expression?

Some common examples of checking if an interval value satisfies an expression include finding the roots of a quadratic equation within a given interval, determining if a given value satisfies a linear inequality, and checking if a value satisfies a trigonometric identity within a specific interval.

What types of mathematical expressions can be checked using interval values?

Interval values can be used to check a wide range of mathematical expressions, including algebraic equations, inequalities, trigonometric expressions, and logarithmic and exponential functions. Essentially, any mathematical expression that involves variables and can be evaluated for a given value can be checked using interval values.

Is it important to check if an interval value satisfies an expression?

Yes, it is important to check if an interval value satisfies an expression because it allows us to determine the validity of our calculations and solutions. It also helps us to identify any potential errors or inaccuracies in our work. Additionally, checking if an interval value satisfies an expression is a fundamental step in solving many mathematical problems and can lead to a better understanding of the concepts involved.

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