Question

Consider the function ln x/ x ⁶. For this function, there are two important intervals: (A, B] and [B,[infinity]) where A and B are critical numbers or numbers where the function is undefined. Find A Find B. For each of the following intervals, tell whether f(x) is increasing or decreasing.

(A, B] _____

Answer 1

`\(A = 1\) and \(B = e^{(1)/(6)}\). On the interval \((A, B]\), \(f(x)\) is increasing.`

Step-by-step explanation:

The critical points of the function
`\(f(x) = (\ln x)/(x^6)\)`occur where the numerator or denominator equals zero or is undefined. In this case, the function is undefined at
`\(x = 0\)` due to the natural logarithm, and the denominator becomes zero at
`\(x = 1\). Thus, \(A = 1\)` is a critical point.

To find B, set the denominator equal to zero and solve for x:

`\[x^6 = e^{(1)/(6)} \implies x = e^{(1)/(6)}.\]`

So,
`\(B = e^{(1)/(6)}\). Now, to determine the behavior of \(f(x)\)` on the interval
`\((A, B]\), we can analyze the sign of the derivative. Calculate \(f'(x)\)` using the quotient rule:

`\[f'(x) = (1 - 6\ln x)/(x^7).\]`

On the interval
`\((A, B]\), \(f'(x)\) is positive because \(\ln x < 1\) for \(x > 1\).`Thus,
`\(f(x)\) is increasing on \((A, B]\).`