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math 1111 - 08 > assignments > test 2 (chapter 3)
test 2 (chapter 3)
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current score 3/15 points
current learning objective: determining whether a function is one - to - one
question 4
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given function (f(x)=5x^{p}-7), which integer values of (p) in range (3,6) make the function one - to - one. answer all values, separated by commas. example: 1,2,3.
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Explanation:

Step1: Recall one - to - one function property

A polynomial function \(y = ax^{n}+bx^{n - 1}+\cdots+z\) is one - to - one if it is either strictly increasing or strictly decreasing. For a function \(f(x)=5x^{p}-7\), its derivative is \(f^\prime(x)=5px^{p - 1}\). When \(p\) is odd, the sign of \(f^\prime(x)\) depends only on the sign of \(x\) for non - zero \(x\) and \(f(x)\) is monotonic.

Step2: Check values of \(p\) in the given range

The integer values of \(p\) in the range \([3,6]\) are \(3\), \(4\), \(5\), \(6\). Among these, \(p = 3\) and \(p = 5\) are odd. When \(p = 3\), \(f(x)=5x^{3}-7\) and \(f^\prime(x)=15x^{2}\geq0\) (and \(f^\prime(x) = 0\) only at \(x = 0\)), and the function is increasing. When \(p = 5\), \(f(x)=5x^{5}-7\) and \(f^\prime(x)=25x^{4}\geq0\) (and \(f^\prime(x)=0\) only at \(x = 0\)), and the function is increasing. When \(p = 4\) or \(p = 6\), the function is an even - degree polynomial and is not one - to - one (it has a minimum or maximum and is symmetric about the y - axis).

Answer:

3,5