Minimum slope of a line tangent to the curve HELP PLEASE

Question

Find the minimum slope of the line tangent to the curve y = x³-3x²+2x+9

Answer 1

y = x³ - 3x² + 2x + 9

Slope of tangent = y'

To minimize y', first take the derivative of y' and set it equal to zero.

(y')' = y" = 6x - 6 = 0

So, x = 1.

When x < 1, (y')' < 0. So, y' is decreasing.

When x > 1, (y')' > 0. So, y' is increasing.

Therefore, y' is minimized when x = 1.

Minimum slope of tangent = y'(1) = 3(1)² - 6(1) + 2 = -1