Question

For the following exercises, consider the graph of the function f and determine where the function is continuous/ discontinuous and differentiable/not differentiable.

Answer 1

Consider the line tangent to the function:

y = 4x - 9

 

Since the line y = 4x - 9 is a tangent to the curve f(x) = x2 - kx, there is only one point of intersection between the line and the curve.

 

Determine the x-coordinate of the point of intersection of the curve and the line as follows:

                4x – 9 = x2 – kx

x2 – kx – 4x + 9 = 0

x2 – (k + 4)x + 9 = 0

 

Since there is only one point of intersection, the discriminant of the quadratic equation x2 – (k + 4)x + 9 = 0 should be 0.

 

For a quadratic equation ax2 + bx + c = 0, the value of discriminant (D) is given by the formula as follows:

D = b2 – 4ac

 

Substitute a = 1, b = -(k + 4), c = 9 to find the discriminant of the quadratic equation x2 – (k + 4)x + 9 = 0 which is equal to 0.

(k + 4)2 – 4 × 9 = 0

            (k + 4)2 = 36

                k + 4 = ± 6

                      k = 2, -10

 

Therefore, the given line y = 4x – 9 is tangent to the curve f(x) = x2 - kx for k = 2 and k = -10.

The given line y = 4x – 9 is tangent to the curve f(x) = x2 - kx for k = 2 and k = -10.