Questions: Let f(x)=x^2-10x-5. a. Find the values of x for which the slope of the curve y=f(x) is 0. b. Find the values of x for which the slope of the curve y=f(x) is -2.

Transcript text: Let $f(x)=x^{2}-10 x-5$. a. Find the values of $x$ for which the slope of the curve $y=f(x)$ is 0 . b. Find the values of $x$ for which the slope of the curve $y=f(x)$ is -2 .

Solution

Solution Steps

To solve the given problem, we need to find the values of $ x $ for which the slope of the curve $ y = f(x) $ is 0 and -2. The slope of the curve at any point is given by the derivative of $ f(x) $.

Compute the derivative of $ f(x) $.
Set the derivative equal to 0 and solve for $ x $.
Set the derivative equal to -2 and solve for $ x $.

Step 1: Find the Derivative

The function is given by $ f(x) = x^2 - 10x - 5 $. To find the slope of the curve, we first compute the derivative: \[ f'(x) = 2x - 10 \]

Step 2: Solve for Slope Equal to Zero

To find the values of $ x $ for which the slope is 0, we set the derivative equal to 0: \[ 2x - 10 = 0 \] Solving for $ x $ gives: \[ 2x = 10 \implies x = 5 \]

Step 3: Solve for Slope Equal to Negative Two

Next, we find the values of $ x $ for which the slope is -2 by setting the derivative equal to -2: \[ 2x - 10 = -2 \] Solving for $ x $ gives: \[ 2x = 8 \implies x = 4 \]