NATIONAL UNIVERSITY OF SINGAPORE
MA2002 Calculus
Homework 3
This homework is due 23:59 on 7th October 2024.
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1. (1 mark) Suppose f (x) is a differential function on R, such that f ′ (x) ≤ 1 for all 1 ≤ x ≤ 4.
Show that f (4) − f (1) ≤ 3.
2. (1 marks) Find the extreme values of the function f (x) = x 4/3 on the interval [−1, 8].
3. (2 marks) Let f (x) = −x 3 + 2x 2 . Find the intervals on which the function is increasing and
decreasing; identify the function’s local and absolute extreme values, if any, say where they
occur.
4. (2 marks) Sketch the graph of a function f that is differentiable on R such that
f (−2) = 8,
f ′ (2) = f ′ (−2) = 0,
f (0) = 4,
f ′ (x) < 0 for |x| < 2,
f (2) = 0,
f ′′ (x) < 0 for x < 0,
f ′ (x) > 0 for |x| > 2,
f ′′ (x) > 0 for x > 0.
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