Parametric Inequality: Find b to Solve x^4-2x^2<a^2-1

[solakis1]

yesterday i come across the following inequality;

Given a>0 find a b>o such that ;

If \(\displaystyle 2-b<x<\frac{1}{4}+b\),then \(\displaystyle x^4-2x^2<a^2-1\)

Can anybody help where to start from??

 

[jvicens]

Hello,

To solve this inequality, we can use algebraic manipulation and substitution. First, let's rewrite the given inequality as:

2-b < x < 1/4 + b

Next, let's focus on the right side of the inequality. We can substitute x with 2 in the expression x^4 - 2x^2 to get:

2^4 - 2(2)^2 = 12

Now, we can substitute a^2 - 1 with 12 in the inequality to get:

2-b < x < 1/4 + b

2-b < x < 1/4 + b

Next, we can simplify the right side of the inequality by finding a common denominator:

2-b < x < \frac{1}{4} + b

2-b < x < \frac{1+4b}{4}

Now, we can set the two inequalities equal to each other:

2-b = \frac{1+4b}{4}

Solving for b, we get b = \frac{1}{5}

Therefore, for any given a>0, we can choose b to be \frac{1}{5} and the inequality will hold true. I hope this helps. Let me know if you have any further questions.