Final answer:
To find the equation of a tangent line to the curve y = 3x² - 9x + 1 at point (4,13), we calculate the derivative, find the slope at x = 4, and then use the point-slope form. Option C is correct.
Step-by-step explanation:
To find the equation of the tangent line to the curve y = 3x² - 9x + 1 at the point (4,13), we first need to determine the derivative of y with respect to x to find the slope of the tangent. The derivative of y, or dy/dx, is obtained by differentiating y = 3x² - 9x + 1 with respect to x, which gives dy/dx = 6x - 9. At x = 4, we substitute this value into the derivative to get the slope of the tangent: 6(4) - 9 = 24 - 9 = 15.
Now, we can use the point-slope form of the equation for a line, y - y1 = m(x - x1), where m is the slope and (x1, y1) is the point of tangency. With the slope m = 15 and point (4,13), the equation becomes y - 13 = 15(x - 4). Simplifying this gives the equation of the tangent line as y = 14x - 25. However, this is not among the provided options, indicating there may be a calculation error or typo in the question or options.
The correct equation of the tangent line is y = 14x - 25.