Maxima And Minima

We are asked to find the maximum slope of the curve defined by the equation:

Step 1: Find the slope function.

The slope of the curve at any point is given by the first derivative of with respect to , denoted as . Let's call the slope function .

So, the slope of the curve at any point is given by the quadratic function .

Step 2: Find the maximum value of the slope function .

The function is a quadratic function representing a parabola. Since the coefficient of the term ( ) is negative, the parabola opens downwards, and thus it has a maximum value at its vertex.

The x-coordinate of the vertex of a parabola occurs at .

In our slope function , and . The value of where the slope is maximum is:

= = =

Alternatively, we can find the maximum by finding the critical points of . We differentiate with respect to and set the derivative to zero.

Set :

To confirm this is a maximum, we check the second derivative :

Since , the slope has a maximum value at .

Step 3: Calculate the maximum slope value.

Substitute into the slope function :

The maximum slope of the curve is 5.

So, the correct answer is (C): 5.