Question

Find all values of x (if any) where the tangent line to the graph of the given equation is horizontal. HINT [The tangent line is horizontal when its slope is zero.] (If an answer does not exist, enter DNE. Enter your answers as a comma-separated list.) y = −9x2 − 2x

Answer 1

By "y = −9x2 − 2x" I assume you meant  y = −9x^2 − 2x (the "^" symbol represents exponentiation).

Let's find the first derivative of y with respect to x:  dy/dx = -18x - 2.  This is equivalent to the slope of the tangent line to the (parabolic) curve.  Now let this derivative (slope) = 0 and solve for the critical value:  -18x - 2 = 0, or
-18x = 2.  Solving for x,   x = -2/18,    or    x = -1/9.

When x = -1/9, y = -9(-1/9)^2 - 2(-1/9).  This simplifies to y = -9/9 + 2/9, or 
y = -7/9.

The only point at which the tangent to the curve is horiz. is (-1/9,-7/9).