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7. Let f be continuous on the interval [3, 5] and differentiable on the interval (3,5), and f(3) = -1, and f(5) = 1. Which of the following theorems implies that the graph of f has a tangent line with a slope , on the interval (3,5)? (a) The Intermediate Value Theorem (b) The Extreme Value Theorem (c) The Mean Value Theorem (d) Rolle's Theorem (e) No theorem guarantees this because the statement is false. 8. Find all critical numbers of the function /(a) = x - 5x/. Classify whether each correspond to a local minimum, local maximum, or neither. 9. Give an example of a function f such that f(5) is a local maximum and f"(5) = 0. 10. Give an example of a function f such that f"(2) is undefined but f(2) is a local minimum. 11. Give an example of a function f such that f"(3) = 0 but f is concave up for all a. 12. Give an example of a function f such that f(3) is undefined, but f(3) is neither a local minimum nor local maximum. 13. Find the linearization, L(x), of the function /(x) = var + 1 at a = 1. 14. Find all critical points of the function f(x) = r(1 - x)$.3. Find all intervals where the function f(x) = 2 - 15r + 10 is increasing. 4. The function f is defined by f(x) = x - 2 Find the x-coordinates of all local extreme values of f. 5. The function f is continuous everywhere and f"(x) = (x - 1)*(3-r)(x + 1)3. Find all intervals where the graph of f is concave down. 6. Find the absolute maximum value, M, and the absolute minimum value, m, of the function f(x) = 1+ = on [1, 4). 7. Let f be continuous on the interval [3,5] and differentiable on the interval (3,5), and f(3) = -1, and f(5) = 4. Which of the following theorems implies that the graph of f has a tangent line with a slope , on the interval (3,5)? (a) The Intermediate Value Theorem (b) The Extreme Value Theorem (c) The Mean Value Theorem (d) Rolle's Theorem (e) No theorem guarantees this because the statement is false. 8. Find all critical numbers of the function f(x) = x - 5x/. Classify whether cach correspond to a local minimum, local maximum, or neither