5–7. For each function ƒ and interval [a, b], a graph of ƒ is given along with the secant line that passes though the graph of ƒ at x = a and x = b.




a. Use the graph to make a conjecture about the value(s) of c satisfying the equation (ƒ(b) - ƒ(a)) / (b-a) = ƒ' (c) .


b. Verify your answer to part (a) by solving the equation (ƒ(b) - ƒ(a)) / (b-a) = ƒ' (c) for c.




ƒ(x) = x² / 4 + 1 ; [ -2, 4] <IMAGE>

Verified step by step guidance

1. Identify the function f(x) = x² / 4 + 1 and the interval [a, b] = [-2, 4].
2. Calculate the values of f(a) and f(b) by substituting a and b into the function: f(-2) and f(4).
3. Use the values obtained to compute the slope of the secant line using the formula (f(b) - f(a)) / (b - a).
4. Set up the equation (f(b) - f(a)) / (b - a) = f'(c) and find the derivative f'(x) of the function f(x).
5. Solve the equation for c by substituting the expression for f'(c) into the equation and finding the values of c that satisfy it.