Question

The tangent to the graph of y=2x³+4x² has a negative gradient when x = k.

Complete the inequality for k.

………..

Answer 1

The inequality for k is 3k² + 4k < 0 when the gradient of y = 2x³ + 4x² is negative at x = k

How to determine the inequality for k.

From the question, we have the following parameters that can be used in our computation:

y = 2x³ + 4x²

The gradient of the equation is the derivative

So, we have

y' = 6x² + 8x

When x = k, the gradient is negative

So, we have

y' < 0

Substitute the known values into the equation

6x² + 8x < 0

Divide through by 2

3x² + 4x < 0

Recall that

x = k

So, we have

3k² + 4k < 0

Hence, the inequality for k is 3k² + 4k < 0