Final answer:
To find the equation of the tangent line to the graph of F(X) = (X-8)(6X+7) at X = 6, we take the derivative of the function to find the slope, then use the point-slope form of a linear equation.
Step-by-step explanation:
To find the equation of the tangent line to the graph of F(X) = (X-8)(6X+7) at X = 6, we need to find the slope of the tangent line at that point and use the point-slope form of a linear equation.
To find the slope, we take the derivative of the function F(X) and evaluate it at X = 6. The derivative of F(X) = (X-8)(6X+7) is 6(6X+7) + (X-8)(6), which simplifies to 42X + 42. When evaluated at X = 6, we get a slope of 270.
Now we can use the point-slope form, which is Y - y = m(X - x), where (x, y) is the given point and m is the slope. Plugging in X = 6 and Y = F(6) into the equation, we get Y - F(6) = 270(X - 6). Simplifying further gives us Y - F(6) = 270X - 1620. This is the equation of the tangent line.