Questions: Find the derivative of the given function. f(x)=(7-8 x)/(1+2 x) f'(x)=-(22)/(1+2 x)^2 Write all x-values (if any) at which the tangent line to the graph would be horizontal. x=

Find the derivative of the given function. f(x)=(7-8 x)/(1+2 x) f'(x)=-(22)/(1+2 x)^2 Write all x-values (if any) at which the tangent line to the graph would be horizontal. x=
Transcript text: Find the derivative of the given function. \[ \begin{array}{r} f(x)=\frac{7-8 x}{1+2 x} \\ f^{\prime}(x)=-\frac{22}{(1+2 x)^{2}} \end{array} \] Write all $x$-values (if any) at which the tangent line to the graph would be horizontal. \[ x= \]
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Solution

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Solution Steps

To find the x-values where the tangent line to the graph is horizontal, we need to determine where the derivative of the function is equal to zero. A horizontal tangent line occurs when the slope of the tangent (the derivative) is zero. Therefore, we set the derivative \( f'(x) \) equal to zero and solve for \( x \).

Step 1: Identify the Condition for Horizontal Tangent

To find the x-values where the tangent line to the graph is horizontal, we need to set the derivative of the function equal to zero. A horizontal tangent occurs when the slope of the tangent line, given by the derivative \( f'(x) \), is zero.

Step 2: Analyze the Derivative

The derivative of the function is given by: \[ f'(x) = -\frac{22}{(1 + 2x)^2} \] We set this equal to zero to find the x-values where the tangent is horizontal: \[ -\frac{22}{(1 + 2x)^2} = 0 \]

Step 3: Solve the Equation

The equation \(-\frac{22}{(1 + 2x)^2} = 0\) has no solutions because a fraction is zero only when its numerator is zero. Since the numerator \(-22\) is a constant and not zero, the equation has no solutions.

Final Answer

\(\boxed{\text{No solutions}}\)

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