Mathematics 53 2nd Semester, A.Y. 2014-2015Exercises 15 - Absolute Extrema Q3, R3, W8, X8

Do the following.

1. TRUE or FALSE

(a) If limx→−2

f(x) = limx→1

f(x) = +∞, then f has no absolute maximum on (−2, 1).

(b) If f has an absolute maximum at x = 2, then f has a relative minimum at x = 2.

(c) Let y = f(x) be continuous on [a, b] and c ∈ (a, b) such that f ′(c) = 0. The y = f9x)has an absolute extremum at x = c.

(d) If the absolute maximum and absolute minimum of f on the interval [a, b] is zero,then f is constant on [a, b].

(e) If f is differentiable on [0, 1], then f has absolute extrema on [0, 1].

(f) If f has absolute minimum and maximum values on [a, b], then f is continuous on[a, b].

(g) The linear function y = mx + b where m 6= 0, has no absolute maximum value onany open interval.

(h) The quadratic function y = ax2 + bx + c where a > 0, has an absolute minimum onR at x = − b

2a.

2. Find the absolute extrema of f(x) = x3 − 6x2 + 9x + 4 on the interval [−1, 2].

3. Find the avsolute extrema of f(x) = 2x3 − 3x2 − 12x + 6 on the interval [−2, 1].

4. Let g(x) =

{(x− 1)2, if x < 3

−5x + 19, if x ≥ 3.

(a) Justify why g has an absolute extreme values on the interval [0, 4]. (Hint: Just checkfor continuity, and apply the Extreme Value Theorem.)

(b) Find the absolute maximum and minimum values of g in [0, 4].

5. Determine the absolute maximum and minimum values of f(x) = −x3 +3x2−2 on (1, 3).

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