5. Find the absolute maximum value and the absolute minimum value of the following
functions in the given intervals:
(i) f (x) = x ^ 3, x \epsilon [- 2, 2]

Answers (1)

Given function is
f(x) = x^3
f^{'}(x) = 3x^2\\ f^{'}(x) = 0\\ 3x^2 = 0\Rightarrow x = 0
Hence, 0 is the critical point of the function f(x) = x^3
Now, we need to see the value of the function f(x) = x^3 at x = 0 and as x \ \epsilon \ [-2,2] we also need to check the value at end points of given range i.e. x = 2 and x = -2
f(0) = (0)^3 = 0\\ f(2= (2)^3 = 8\\ f(-2)= (-2)^3 = -8
Hence, maximum value of function f(x) = x^3 occurs at x = 2 and value is 8
and minimum value of function f(x) = x^3 occurs at x = -2 and value is -8

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