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Explain the key points and reasons for the establishment of 12.3.2(integral Test)
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- Determine if the following statements are true or false. If it is false, explain why?3. Let f: [-2, 2] → R be defined by I f(x):= ²+1' *€ [−2,2]. Find the intervals in which f is either increasing or decreasing. Find the maxima and minima of f.Suppose f is a continuous function on [0, ox) and that | f(t)dt converges to some positive real number I. 00 Define a function G(a) with domain (0, 0) by G(x) = | f(t)dt for every a > 0. Which of the following statements must be true? (Select all true statements.) d G(x) is differentiable on (0, oo) with G(x) = f(x) for all æ > 0. dx | f(t)dt is positive for all æ E (0, o0). There exists N ER with the property that G(x) > 0 for all > N. G(x)3. Define F (s) = [₂ LVI-2²dx, - x² dx, For this function on [-2, 2], find critical points local maxima and minima intervals on which F is increasing and decreasing intervals on which F is concave up or down for s = [-2,2]. You do not need to calculate the value of the function at any point. You may assume without proof that the cube root function is continuous but you do need to refer precisely to any other theorems you are using..5 10) Determine whether the improper integral diverges or converges: 1 dx Vx-1A college professor always finishes his lectures within 2 minutes after the bell rings to end the period and the end of the lecture. Let X = the time (in minutes) that elapses between the bell and the end of the lecture and suppose the pdf of X is: f(x) = (0 x² for 0 ≤ x ≤2 otherwise Compute the exact probability (using integral calculus) that the lecture continues for at least 60 seconds beyond the bell.