The line y = -6x-10 is tangent to the graph of the function y = x ^ 3 + 4x ^ 2-6x-10.

The line y = -6x-10 is tangent to the graph of the function y = x ^ 3 + 4x ^ 2-6x-10. Find the abscissa of the touch point.

y = kx + b – equation of a straight line;

y = -6x – 10; (1)

k = -6;

The slope of the tangent is equal to the tangent of the angle of inclination and is equal to the derivative at the point of tangency:

y ‘= tgх = k;

y ‘= -6;

y ‘= 3x ^ 2 + 8x – 6;

We equate and solve the equation:

3x ^ 2 + 8x – 6 = -6;

3x ^ 2 + 8x = 0;

x (3x + 8) = 0;

x = 0 or x = -8/3;

Substitute the obtained abscissas into the original function and into (1), choose the abscissa in which they are equal:

0 ^ 3 + 4 * 0 ^ 2 – 6 * 0 – 10 = -10;

-6 * 0 – 10 = -10;

Analytically, we see that at -8/3 they are not equal.

Answer: 0.