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Sum

**Determine the maximum and minimum value of the following function.**

f(x) = `x^2 + 16/x`

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#### Solution

f(x) = `x^2 + 16/x`

∴ f'(x) = `2x - 16/x^2`

and f"(x) = `2 + 32/x^3`

Consider, f'(x) = 0

∴ `2x - 16/x^2` = 0

∴ 2x = `16/x^2`

∴ x^{3} = 8

∴ x = 2

The maximum value is 2.

f(x) = `x^2 + 16/x`

For x = 2

f''(2) = `2 + 32/2^3`

= `2 + 32/8`

= 2 + 4

= 6 > 0

∴ f(x) attains minimum value at x = 2

∴ Minimum value = f(2) = `(2)^2 + 16/2 = 4 + 8` = 12

∴ The function f(x) has minimum value 12 at x = 2.

Concept: Maxima and Minima

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