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Sivakumar MATH 171 Exercise Set 9 1. (i) Give an example of a function f and a point c such that c belongs to the domain of f , f 0 (c) = 0, but f does not have a local extremum at c. (ii) Give an example of a function f and a point c such that f is continuous at c, f 0 (c) does not exist, but f does not have a local extremum at c. Note that in both cases above, c is a critical number for f . The next two examples are slightly modified versions of questions from an old exam (1989) I gave at the University of Alberta, my alma mater: 2. Suppose f is continuous on [a, b] and |f 0 (x)| ≤ 1 for every x in (a, b). Show that |f (v)−f (u)| ≤ |v − u| for every a ≤ u < v ≤ b. sin x , 0 < x ≤ π, and f (0) = 1. Let g(x) = x cos x − sin x, 0 ≤ x ≤ π. x (i) Observe that f and g are continuous on [0, π]. The continuity of f at 0 requires some thought; everything else is quite clear. 3. Let f (x) = (ii) Find g 0 (x), 0 < x < π, and use it to show that g is decreasing on [0, π]. (iii) Evaluate g(0) and use it, along with (ii), to show that g(x) < 0, for every 0 < x ≤ π. g(x) , 0 < x < π. x2 (v) Use (iv) and (iii) to show that f is decreasing on [0, π]. sin a a (vi) Suppose 0 < a < b < π. Show that > . sin b b (iv) Show that f 0 (x) = The function f considered in Example 3 above is called the sinc function. It plays a useful rò‚le in Signal Analysis, in particular, the Theory of Sampling. 4. Show that the equation x101 + x51 + x − 1 = 0 has exactly one real root (that is, there exists exactly one real number which satisfies the said equation). Do this in two steps: firstly, show that there is at least one such root, and, secondly, that there cannot be more than one. 5. A number c is called a fixed point of a function f if f (c) = c. Suppose f is a differentiable function and f 0 (x) < 1 for all x. Show that f has at most one fixed point. (That is, show that f cannot have two or more fixed points.) 6. (from a high-school exam given in Madras, India; I happened to come upon this test some years ago.) Find all possible values of the real number a such that the function f (x) = x3 + ax2 + 48x + 12 is always increasing (that is, increasing on (−∞, ∞)). 7. Determine all possible values of the number b for which the function f (x) = (b2 + b − 6) cos 2x + (b − 2)x + cos 1 has no critical number. 8. (i) Show that tan x ≥ x for every 0 ≤ x < π/2. x ≤ tan−1 x ≤ x for every x ≥ 0. (ii) Show that 1 + x2 1