Page 347 - DMTH201_Basic Mathematics-1

Basic Mathematics – I

Notes d A 4
2
Further, 0 . Hence second order condition for maxima of A is also satisfied.
dx 2
13.3 Summary

Let f(x) be a function with domain D. Then f(x) has an absolute maxima at a point c in D if
f(x) f(c) for all x in D and an absolute minima at a point d in D if f(x) f(d) for all x in D.
Absolute maxima/minima are also called global maxima/minima.
A function f(x) has a local maxima (or minima) at an interior point c in its domain D if f(x)
f(c) (or f(x) f(c)) for all x in some open interval containing c.
If a function f(x) has a local extrema (i.e., maxima or minima) at an interior point c of its
domain, and if f (c) exists, then f (c) = 0.

When the function f(x) is twice differentiable at an interior point c of the domain, then
f(x) has a local maxima at x = c if f (c) = 0 and f (c) < 0.
f(x) has a local minima at x = c if f (c) = 0 and f (c) > 0.
When f(x) has a maxima (or minima) at c, the curve of f(x) is concave (or convex) from
below. This test is inconclusive when f (c) = 0.

13.4 Keywords

Absolute Maxima/Minima (Definition): Let f(x) be a function with domain D. Then f(x) has an
absolute maxima at a point c in D if f(x) f(c) for all x in D and an absolute minima at a point d
in D if f(x) f(d) for all x in D. Absolute maxima/minima are also called global maxima/minima.
Local Maxima/Minima (Definition): A function f(x) has a local maxima (or minima) at an interior
point c in its domain D if f(x) f(c) (or f(x) f(c)) for all x in some open interval containing c.

13.5 Self Assessment

2
1. Determine maxima of y x 3 (x 1)

2 5
(a) (b)
5 2
2 2
(c) (d)
3 6
5
2. Find maximum value of y if y x 2 2x 3,x then y is equal to:
2
(a) 110.75 (b) 119.12

(c) 118.75 (d) 111.85
2
2
3. If f(x) = x 4x 3x + x then find f(x), if x = 4
(a) 2 (b) 16
(c) 10 (d) 10

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