Questions: Find the absolute extrema if they exist, as well as all values of x where they occur, for the function f(x)=(6-x)/(5+x) on the domain [0,5]. Find the derivative of f(x)=(6-x)/(5+x).

Find the absolute extrema if they exist, as well as all values of x where they occur, for the function f(x)=(6-x)/(5+x) on the domain [0,5]. Find the derivative of f(x)=(6-x)/(5+x).
Transcript text: Find the absolute extrema if they exist, as well as all values of $x$ where they occur, for the function $f(x)=\frac{6-x}{5+x}$ on the domain [0,5]. Find the derivative of $f(x)=\frac{6-x}{5+x}$. \[ f^{\prime}(x)= \] $\square$
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Solution

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Solution Steps

Step 1: Find the Derivative

The derivative of the function \( f(x) = \frac{6 - x}{5 + x} \) is calculated using the quotient rule. The result is: \[ f'(x) = -\frac{6 - x}{(5 + x)^2} - \frac{1}{5 + x} \]

Step 2: Identify Critical Points

Setting the derivative \( f'(x) \) equal to zero to find critical points yields no solutions in the interval [0, 5]. Thus, there are no critical points.

Step 3: Evaluate the Function at Endpoints

Next, we evaluate the function at the endpoints of the interval:

  • At \( x = 0 \): \[ f(0) = \frac{6 - 0}{5 + 0} = \frac{6}{5} \]
  • At \( x = 5 \): \[ f(5) = \frac{6 - 5}{5 + 5} = \frac{1}{10} \]
Step 4: Determine Absolute Extrema

The values of the function at the endpoints are:

  • \( f(0) = \frac{6}{5} \)
  • \( f(5) = \frac{1}{10} \)

The absolute maximum value is \( \frac{6}{5} \) at \( x = 0 \), and the absolute minimum value is \( \frac{1}{10} \) at \( x = 5 \).

Final Answer

The absolute extrema of the function \( f(x) = \frac{6 - x}{5 + x} \) on the interval [0, 5] are:

  • Absolute maximum: \( \boxed{\frac{6}{5}} \) at \( x = 0 \)
  • Absolute minimum: \( \boxed{\frac{1}{10}} \) at \( x = 5 \)
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