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