Extreme values, the Mean Value Theorem, curve sketching and concavityChapter 6: Practice/review problemsThe collection of problems listed below contains questions taken from previous MA123 exams.Extreme values problems on a closed interval[1]. Suppose f(t) = √4 − t if t < 4√t − 4 if t ≥ 4.Find the minimum of f (t) on the interval [0, 6].(a) 0 (b) 2 (c) 4 (d) 6 (e) 8[2]. Let g(s) =s − 1s + 1. Find the maximum of g(s) on the interval [0, 2].(a) −1/3 (b) 0 (c) 1/3 (d) 2/3(e) Neither th e maximum nor the min imum exists on the given interval.[3]. Suppose f(t) =t2− 2t + 2 if t < 1t3if t ≥ 1.Find the minimum of f (t) on the interval [0, 2].(a) −1 (b) 0 (c) 1 (d) 2 (e) 8[4]. Let f(x) = 3x2+ 6x + 4. Find the maximu m value of f (x) on the interval [−2, 1].(a) 5 (b) 7 (c) 9 (d) 13 (e) −1[5]. Let G(x) =((x −3) + 6 if x ≥ 3−(x − 3) + 6 if x < 3.Find the minimum of G(x) on the interval [−10, 10].(a) 3 (b) 1 (c) −6 (d) 19 (e) 6[6]. Let g(s) =1s + 1. Find the maximum of g(s) on the interval [0, 2].(a) −1 (b) 0 (c) 1 (d) 2(e) Neither th e maximum nor the min imum exists on the given interval.[7]. Find the minimum value of f(x) = x3− 3x + 3 on the interval [−2, 4].(a) 2 (b) 1 (c) 0 (d) −1 (e) −289[8]. Find the maximum of g(t) = |t + 4|+ 10 on the interval [−12, 12].(a) 19 (b) 20 (c) 24 (d) 26 (e) 28[9]. Find the minimum value of f(x) =√x2− 2x + 16 on the interval [0, 5].(a) 1 (b) 2(c)√15(d)√12(e) 0[10]. Let f (x) = |x2− 1| + 2. Find the minimum of f (x) on the interval [−3, 3].(a) 3 (b) 0 (c) 1 (d) 2 (e) −1[11]. Suppose f(t) = 2t3− 9t2+ 12t + 31. Find the value of t in the interval [0, 3] where f (t) takes on itsminimum.(a) 0 (b) 1 (c) 2 (d) 3(e) Neither th e maximum nor the min imum exists on the given interval.[12]. Let Q(t) = t2. Find a value A such that the average rate of change of Q(t) from 1 to A equals theinstantaneous rate of change of Q(t) at t = 2A(a) 1(b)13(c)14(d)15(e) Does not existMean Value Theorem problems[13]. Find the value of A su ch that the average rate of change of the function g(s) = s3on the interval [0, A]is equal to the ins tantaneous rate of change of the function at s = 1.(a)√2 (b)√3 (c)√5 (d)√6 (e)√12[14]. Suppose k(s) = s2+ 3s + 1. Find a value c in the interval [1, 3] such th at k′(c) equals the average rate ofchange of k(s) on the interval [1, 3].(a) −1 (b) 0 (c) 1 (d) 2 (e) 3[15]. Let k(x) = x3+ 2x. Find a value of c between 1 and 3 such that the average r ate of change of k(x) fromx = 1 to x = 3 is equal to the instantaneous rate of change of k(x) at x = c.(a) 30 (b) 15(c)r283(d)r133(e) 60Increasing/decreasing problems[16]. Which function is always increasing on (0, 2)(a)√x + x2(b) x + (1/x)(c) x3− 3x(d) 7 − |x|(e) (x −1)490[17]. Suppose that a function f (x) has derivative f′(x) = x2+ 1. Which of the following statements is trueabout the graph of y = f(x)?(a) The function is increasing on (−∞, ∞)(b) The function is decreasing on (−∞, ∞)(c) The function is increasing on (−∞, −1) and (1, ∞), and the fun ction is decreasing on (−1, 1).(d) The function is increasing on (−∞, 0), and the function is decreasing on (0, ∞).(e) The function is decreasing on (−∞, 0), and the function is increasing on (0, ∞).[18]. Find the largest value of A such that the function g(s) = s3− 3s2− 24s + 1 is increasing on the interval(−5, A).(a) −4 (b) −2 (c) 0 (d) 2 (e) 4[19]. Let f (x) = e−x2. Find the intervals where f(x) is decreasing.(a) (−∞, 0) (b) (0, ∞) (c) (−∞, −1)(d) (1, ∞) (e) (−1, 1)[20]. Let f (x) = x ln x. Find the intervals where f(x) is increasing.(a) (0, ∞) (b) (1, ∞) (c) (e, ∞)(d) (1/e, ∞) (e) (1/e, e)[21]. Suppose the cost, C(q), of stocking a quantity q of a product equals C(q) =100q+ q. The rate of changeof the cost with respect to q is called the marginal cost. When is the marginal cost positive?(a) q > 10 (b) q > 15 (c) q < 20 (d) q < 25 (e) q = 30[22]. For which values of t is the fun ction t3− 2t + 1 increasing?(a) t >p2/3 or t < −p2/3 (b) −p2/3 < t <p2/3 (c) 0 < t <p4/3(d) −p4/3 < t < 0(e) Never[23]. Suppose that g′(x) = x2− x − 6. Find the interval(s) where g(x) is increasing.(a) (−1, 2) (b) (−∞, −2) and (3, ∞) (c) (−∞, −1) and (2, ∞)(d) (−2, 3) (e) It cannot be determined from the information given[24]. Let f (x) = xe2x. Then f is decreasing on the following interval.(a) (−∞, −1/2) (b) (−1/2, ∞) (c) (−∞, 1/2)(d) (1/2, ∞) (e) (−∞, 0)91[25]. Find the interval(s) where f(x) = −x3+ 18x2− 105x + 4 is increasing.(Note that th e coefficient of x3is −1, so compute carefully.)(a) (−∞, 5) and (7, ∞) (b) (5, 7) (c) (−∞, −5) and (7, ∞)(d) (−5, 7) (e) (−7, 5)[26]. Suppose that f(x) = xg(x), and for all positive values of x the function g(x) is negative (i.e., g(x) < 0)and decreasing. Which of the following is true for the function f (x)?(a) f(x) is negative and decreasing for all positive values of x.(b) f(x) is positive and increasing for all positive values of x.(c) f(x) is negative and increasing for all positive values of x.(d) f(x) is positive and decreasing for all positive values of x.(e) None of the above.[27]. Suppose the derivative of a function g(x) is given by g′(x) = x2− 1. Find all intervals on which g(x) isincreasing.(a) (−∞, ∞) (b) (−1, 1) (c) (−∞, −1) and (1, ∞)(d) (0, ∞) (e) (−∞, 0)Extreme values problems using the first derivative[28]. Suppose the derivative of the function h(x) is given by h′(x) = 1 −|x|. Find the value of x in the interval[−1, 1] where h(x) takes on its minimum value.(a) −1/2 (b) −1 (c) 0 (d) 1/2 (e) 1[29]. Suppose the total cost, C(q), of producing a quantity q of a produ ct equalsC(q) = 1000 + q +110q2.The average cost, A(q), equals the total cost divided by the quantity produced. What is the minimumaverage cost? (Assume q > 0)(a) 20 (b) 21 (c) 26 (d) 30 (e) 31[30]. Suppose that a function h(x) has derivative h′(x) = x2+ 4. Find the x value in the interval [−1, 3] whereh(x) takes its minimum.(a) −1 (b) 3 (c) 5 (d) 13 (e) 2992[31]. Suppose the cost, C(q), of stocking a quantity q of a pr oduct equals C(q) …