Final answer:
To determine the values of a, b, and c such that the function f satisfies the hypotheses of the mean value theorem on the interval [0, 3], we need to find the derivative of the function, verify the conditions for the mean value theorem, calculate the average rate of change of the function, and graphically represent the function and its tangent line at the mean value.
Step-by-step explanation:
To determine the values of a, b, and c such that the function f satisfies the hypotheses of the mean value theorem on the interval [0, 3], we need to find the derivative of the function, verify the conditions for the mean value theorem, calculate the average rate of change of the function, and graphically represent the function and its tangent line at the mean value.
a) The derivative of f(x) can be found by taking the derivative of each piece of the function. The derivative of 1 is 0, the derivative of ax + b is a, and the derivative of x^2 + 4x + c is 2x + 4. So the derivative of f(x) is 0 for x = 0, a for 0 < x ≤ 1, and 2x + 4 for 1 < x ≤ 3.
b) To verify the conditions for the mean value theorem, we need to check if f is continuous on [0, 3] and differentiable on (0, 3). The function f is continuous at x = 0, x = 1, and x = 3 since the pieces of the function are continuous. The function f is differentiable at x = 0, the limit of f'(x) as x approaches 0 from the right exists, and the limit of f'(x) as x approaches 0 from the left exists. Similarly, the function f is differentiable at x = 1 and x = 3. Therefore, the conditions for the mean value theorem are satisfied.
c) The average rate of change of a function on an interval [a, b] can be calculated using the formula: average rate of change = (f(b) - f(a))/(b - a). In this case, the average rate of change of f on the interval [0, 3] is (f(3) - f(0))/(3 - 0). Since f(3) = 12 + c and f(0) = 1, the average rate of change is (12 + c - 1)/3.
d) To graphically represent the function and its tangent line at the mean value, you can plot the points (0, 1), (1, a + b), and (3, 9 + 6a + c) and draw a curve that connects them. Then, calculate the derivative at the mean value of x and find the slope of the tangent line. Finally, use the slope-intercept form of a line to draw the tangent line through the point on the curve corresponding to the mean value of x.