12. Consider the function A(x) = sin(t2) dt. (a) Find A'(x) (b) Find the x-value of the first positive critical number of A(x) (c) At that first positive critical number you found in (b), does A(x) have a local max, a local min, or neither? Justify your answer

Use the following information to solve the first two questions. Gamma function r(a) = ²₁-¹₁e" du In particular, for integer a ≥ 1 Gamma function's value is I'(a) = (a - 1)! Beta Function where a>0 and b>0,. B(a,b) = √' uª-¹ . (1 – u)b-¹ du Relation between Gamma function and Beta function -1. (10 points) Evaluate B(a,b) = г(а).г(b) [(a+b) -4x [x².etdx. C 2

(b) Suppose that the function f (x) has the form f(x) : тх + b' where C is a non-zero constant and m + 0. Explain why it is impossible for f(x) to have any local max's or local min's.

4

Find a good numerical approximation to F(4) for the function with the properties that F'(x) = e-x*15 and F(0) = 1.

In x 6. Let f be the function given by f(x) = for all x > 0. The derivative of f is given by f'(x) = (a) Write an equation for the line tangent to the graph of f at x = e“. (b) Find the x-coordinate of the critical point of f. Determine whether this point is a relative minimum, a relative maximum, or neither for the function f. Justify your answer. (c) The graph of the function f has exactly one point of inflection. Find the x-coordinate of this point. (d) Find lim f (x).