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Introduction to Functions

Can there be ...

Question

Can there be two absolute maximums?

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Solution

Explanation for the correct answer:
Although a function can have only one absolute minimum value and only one absolute maximum value (in a specified closed interval),
But absolute maximum or absolute minimum can occur at more than one place in the domain.
Hence, there cannot be two absolute maximums.

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The absolute maximum & minimum values of functions can be found by their monotonic & asymptotic behavior provided they exist. We may agree that finite limiting values may be regarded as absolute maximum or minimum. For example, the absolute maximum value of

\frac{1}{1 + x^{2 m}} (m ϵ N)

x = 0

, on the other side absolute minimum value of the some function is 0, which is limiting value of the function when

x \to - \infty

x \to + \infty

f^{'} (x) = 0

f^{''} (x) = 0

x = a

f^{^{'} "} (x) \neq 0

x = a

, then f(x) is neither absolute maximum nor absolute minimum at

x = a

x = a

is called point of inflexion.

On the basis of above information answer the following questions.

x^{4} - 4 x + 1

Q. 1)Find the absolute maximum value and the absolute minimum value of the function in the given interval:
i)

f (x) = x^{3}, x ϵ [- 2, 2]

2)Find the absolute maximum value and the absolute minimum value of the function in the given interval:
ii)

f (x) = sin x + cos x, x ϵ [0, π]

3)Find the absolute maximum value and the absolute minimum value of the function in the given interval:
iii)

f (x) = 4 x - \frac{1}{2} x^{2}, x ϵ [- 2, \frac{9}{2}]

4)Find the absolute maximum value and the absolute minimum value of the function in the given interval:
iv)

f (x) = (x - 1)^{2} + 3, x ϵ [- 3, 1]

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