Find the abscissas of the points at which the tangents to the curve y = x ^ 3 – 9x ^ 2 + 11x

Find the abscissas of the points at which the tangents to the curve y = x ^ 3 – 9x ^ 2 + 11x + 10 are parallel to the straight line y = 15 – 4x.

Find the points at which the derivative of this function is -4.

At these points, the slopes of the tangents to the graph of this function and the straight line y = 15 – 4x will coincide, therefore, at these points the tangents to the graph of this function will be parallel to the straight line y = 15 – 4x.

y ‘= (x ^ 3 – 9x ^ 2 + 11x + 10)’ = 3x ^ 2 – 2 * 9x + 11 = 3x ^ 2 – 18x + 11;

3x ^ 2 – 18x + 11 = 0;

x = (9 ± √ (81 – 3 * 11)) / 3 = (9 ± √ (81 – 33)) / 3 = (9 ± √48) / 3 = (9 ± 4√3) / 3;

x1 = (9 + 4√3) / 3;

x2 = (9 – 4√3) / 3.

Answer: x = (9 – 4√3) / 3 and x = (9 + 4√3) / 3.