How do you find a tangent line parallel to secant line?

You can find a tangent line parallel to a secant line using the Mean Value Theorem.

The Mean Value Theorem states that if you have a continuous and differentiable function, then

##f'(x) = (f(b) - f(a))/(b - a)##

To use this formula, you need a function ##f(x)##. I'll use ##f(x) = -x^3## as an example.

I'll also use ##a = -2## and ##b = 2## for the interval for the secant line. This is the line that passes through the points ##(-2, 8)## and ##(2, -8)##.

So, we know that the slope of this line will be ##(-8 - 8)/(2 - (-2)) = -4##.

To find the tangent lines parallel to this secant line, we will take the function's derivative, ##f'(x)##, and set it equal to ##-4##, then solve for ##x##.

##-3x^2 = -4##

Solving this for ##x## gives us: ##x = ±sqrt(4/3)##.

So, the lines tangent to ##y = -x^3## at ##x = sqrt(4/3)## and ##x = -sqrt(4/3)## must be parallel to the secant line passing through ##x = 2## and ##x = -2##.