Final answer:
To find an integer c such that d/dx |ₓ₌₁( f/g )=2, we need to find the derivative of f(x)/g(x) at x=1 and set it equal to 2.
Step-by-step explanation:
To find an integer c such that d/dx |ₓ₌₁( f/g )=2, we need to find the derivative of f(x)/g(x) at x=1 and set it equal to 2. First, let's find f(x)/g(x) by substituting the given functions f(x) and g(x) into the expression. f(x) = 1/3x and g(x) = 1/3x - c. Therefore, f(x)/g(x) = (1/3x) / (1/3x - c). Now, to find the derivative at x=1, we need to find the derivative of f(x)/g(x) and evaluate it at x=1.
To find the derivative, we can use the quotient rule: d/dx [f(x)/g(x)] = (g(x)*f'(x) - f(x)*g'(x)) / (g(x))^2. Plugging in f(x) = 1/3x and g(x) = 1/3x - c, we get d/dx [f(x)/g(x)] = [(1/3x - c)*(1/3) - (1/3x)*(1/3)] / [(1/3x - c)^2]. After simplifying this expression, we get d/dx [f(x)/g(x)] = [(1 - 3c)/(9x - 3cx - 3c^2)]. Now, we evaluate this expression at x = 1: d/dx |ₓ₌₁( f/g ) = [(1 - 3c)/(9 - 3c - 3c^2)] = 2. Now, we solve this equation to find the value of c.