Final answer:
The tangent lines for the function f(x) = x² at x = -1 and x = 4 can be found by finding the derivative of the function, evaluating it at the given points, and using the slope-intercept form of a line to write the equations of the tangent lines.
Step-by-step explanation:
The tangent line for the function f(x) = x² at x = -1 and x = 4 can be found using the slope of the function at those points. The slope of a tangent line at a given point is equal to the derivative of the function evaluated at that point.
First, find the derivative of the function f(x) = x². The derivative of x² is 2x.
Next, plug in the values of x = -1 and x = 4 into the derivative 2x to find the slopes at those points. The slope at x = -1 is 2*(-1) = -2 and the slope at x = 4 is 2*4 = 8.
Using the slopes and the corresponding points, we can write the equations of the tangent lines. The tangent line at x = -1 is given by y = (-2)(x - (-1)) + f(-1) and the tangent line at x = 4 is given by y = 8(x - 4) + f(4).