Differentiation question?

Question

Find the coordinates of the two points on the curve with equation x^3-6x+2 where the gradient is 6.

Answer 1

The equation is x³ - 6x + 2 = 0 and the gradient is 6.

The related function is f(x) = x³ - 6x + 2.

Differentiate with respect to x.

f'(x) = 3x² - 6

Therefore, 3x² - 6 = 6

3x² = 12

x² = 4

x = ± 2.

f(2) = 2³ - 6(2) + 2 = -2.

f(-2) = (-2)³ - 6(-2) + 2 = 6.

The points are (2, -2) and (-2, 6).