find the critical numbers of the function. (enter your answers as a comma - separated list. if an answer…

Explanation:

Step1: Find the derivative of the function

Use the power rule ( (x^n)^\prime=nx^{n - 1}). For (g(x)=x^{\frac{1}{7}}-x^{-\frac{6}{7}}), the derivative (g^\prime(x)=\frac{1}{7}x^{\frac{1}{7}-1}-\left(-\frac{6}{7}\right)x^{-\frac{6}{7}-1}) (g^\prime(x)=\frac{1}{7}x^{-\frac{6}{7}}+\frac{6}{7}x^{-\frac{13}{7}}) Factor out (\frac{1}{7}x^{-\frac{13}{7}}): (g^\prime(x)=\frac{1}{7}x^{-\frac{13}{7}}(x + 6))

Step2: Set the derivative equal to zero and find where it is undefined

• Set (g^\prime(x)=0): (\frac{1}{7}x^{-\frac{13}{7}}(x + 6)=0) Since (\frac{1}{7}x^{-\frac{13}{7}}\neq0) for (x\neq0), then (x+6 = 0), which gives (x=-6)
• Find where (g^\prime(x)) is undefined: The derivative (g^\prime(x)=\frac{1}{7}x^{-\frac{6}{7}}+\frac{6}{7}x^{-\frac{13}{7}}=\frac{x + 6}{7x^{\frac{13}{7}}}) is undefined at (x = 0). But the original function (g(x)=x^{\frac{1}{7}}-x^{-\frac{6}{7}}=\sqrt[7]{x}-\frac{1}{x^{\frac{6}{7}}}) is also undefined at (x = 0) (because of the (x^{-\frac{6}{7}}) term). So we discard (x = 0) as it is not in the domain of (g(x))

Answer:

(-6)