Given the function f defined by f(x) = x²e^-2x, x > 0. Then f(x) has the maximum value equal to?
1/e, 1/2e, 1/e², 4/e⁴

Solution:

A function is a process or a relation that associates each element 'a' of a non-empty set A , at least to a single element 'b' of another non-empty set B. 

A relation ‘f’ from a set A (the domain of the function) to another set B (the co-domain of the function) is called a function in math.

f = {(a, b)| for all a ∈ A, b ∈ B}

The function f given is

f (x) = x²e^-2x

Differentiating with respect to x

f’ (x) = e^-2x(2x) - (2x²)e^-2x

By further simplification

f’ (x) = 2e^-2xx(1 - x)

To find the the critical value, put f'(x) = 0 to find

2e^-2xx(1 - x) = 0

⇒ 1-x = 0

⇒ x = 1

Substitute this value in f(x).

f(x) = x²e^-2x 

f(1)= 1 e^-2

⁼1/ e²

Thus f (x) will attain its maximum at x = 1 which is 1 / e².

Therefore, the maximum value is 1 / e².

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Given the function f defined by f(x) = x²e^-2x, x > 0. Then f(x) has the maximum value equal to?

Summary:

Given the function f defined by f(x) = x²e^-2x, x > 0. Then f(x) has the maximum value equal to1 / e².