Question

Let f be a differentiable function with f(1)=-2 and f´(1)=2. Let the function g(x)=(2x-6)/f(x). Write the equation of the line tangent to the graph of g at the point where x=1.​

Answer 1

To find the equation of the line tangent to the graph of g at the point where x=1, you need to find the derivative of g and evaluate it at x=1. Then, use the point-slope form of a linear equation to write the equation of the tangent line using the point (1, g(1)) and the slope found in the previous step.

Step-by-step explanation:

To find the equation of the line tangent to the graph of g at the point where x=1, we need to find the derivative of g and evaluate it at x=1.

Given the function g(x) = (2x-6)/f(x), we need to find the derivative of g(x) by using the quotient rule. The derivative of g(x) is g'(x) = (f(x)*(2) - (2x-6)*f'(x))/(f(x))^2.

Now, substitute x=1 into g'(x) to find the slope of the tangent line at x=1. Finally, use the point-slope form of a linear equation to write the equation of the tangent line using the point (1, g(1)) and the slope found in the previous step.