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Let , where .Prove that if then cannot take values between and .
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Q. Let , where .Prove that if then cannot take values between and .
- Simplify Expression for : Let's start by simplifying the expression for by performing polynomial division or factoring, if possible.We can try to factor the numerator, but since the roots of are complex, we cannot factor it over the real numbers. Therefore, we will perform polynomial division.
- Perform Polynomial Division: Perform polynomial division of the numerator by the denominator. gives us with a remainder of .So,
- Analyze Expression : Now, let's analyze the expression . We know that cannot be , so the term is always defined. For to be between and , the following inequality must be true:
- Subtract from Inequality: Subtract from all parts of the inequality to isolate the fraction on one side.
- Consider Left Part of Inequality: Now, let's consider the two parts of the inequality separately.First, let's look at the left part: .For , the term is positive, and thus is always greater than . Since can be any real number greater than , there will always be values of for which is greater than .
- Consider Right Part of Inequality: Now, let's look at the right part of the inequality: . For , the term is positive, and thus is always greater than . For to be less than , would have to be negative and its absolute value would have to be greater than . However, as approaches from the right, becomes arbitrarily large, so there is no real number for which is less than .
- Conclusion: Since there is no real number for which is less than , the original inequality cannot be satisfied for any real . Therefore, cannot take values between and for any real .
