Final answer:
To determine the increasing and decreasing intervals of the function f(x) = x^2 - 7x + 6, find the derivative of the function, set it equal to zero to find the critical points, and analyze the sign of the derivative on either side of the critical points. The correct answer is d) Increasing: (-[infinity], 4), Decreasing: (4, [infinity]).
Step-by-step explanation:
To determine the increasing and decreasing intervals of the function f(x) = x^2 - 7x + 6, we need to analyze the slope of the function. The slope of a function can be found by taking the derivative of the function. Taking the derivative of f(x) = x^2 - 7x + 6 gives us f'(x) = 2x - 7.
Now, we need to find the critical points of the function by setting f'(x) = 0. Solving 2x - 7 = 0 gives us x = 7/2.
Since the derivative f'(x) = 2x - 7 is positive for x < 7/2 and negative for x > 7/2, the function f(x) = x^2 - 7x + 6 is increasing for x < 7/2 and decreasing for x > 7/2. Therefore, the correct option is d) Increasing: (-[infinity], 4), Decreasing: (4, [infinity]).