If you could show your work and provide a brief explanation, that would be great. Thank you!

55. Consider the image below that has a graph of a continuous function. and select the statement that is true. A} The Intermediate Value Theorem guarantees that the function has a root at A. B] The Intermediate Value Theorem might guarantee that the funetion has a root at A. C The Intermediate Value Theorem guarantees that the function has a root at E. 1' D} The Intermediate Value Theorem guarantees that the function might have a root at C. E} None of the above. 53. Which theorem is this: Let ts} be continuous over a elosed, bounded interval [a, h]. If 2 is any real number between e] and t], then there is a number :3 in [a, b] satisfying e] = z. A E } Fermat's Theorem 1' C} The Extreme Value Theorem 1' } Rolle's Theorem D The Intermediate Value Theorem E Kone of the above. 57. The First Derivative Test is able to classify a critical point as a local maximum when A) the first derivative is negative on the interval to the left of the critical point, and then positive on the interval to the right of the critical point. B) the first derivative is positive on the interval to the left of the critical point, and then positive on the interval to the right of the critical point. C) the first derivative is positive on the interval to the left of the critical point, and then negative on the interval to the right of the critical point. D) the first derivative is negative on the interval to the left of the critical point, and then positive on the interval to the right of the critical point. E) None of the above. 58. In a population of size 500 at time t = 0, if there are 50 births per year and 25 deaths per year, what is the solution to the differential equation that describes the change in population per year? A) N(t) = 0.050.5t C) N(t) = 500e-0.01t E) None of the above. B) N(t) = 25t D) N(t) = 500e0.05t