Consider a function f(x) = x^3*e^-2x defined for 0. find the maximal value of this function and validate your solution using differentiation techniques.

The maximum value of the function is 0.168 at x=1.5.

Step-by-step explanation:

The given function is

We need to find the maximal value of this function.

Differentiate the given function with respect to x.

Product rule of differentiation:

Using product rule, we get

             .... (1)

Equate f'(x)=0.

The function has two critical values 0 and 1.5.

Differentiate the f'(x) with respect to x.

At x=0,

Since f''(0)=0, therefore x=0 is the point of inflection.

At x=1.5,

Since f''(1.5)<0, therefore the function is maximum at x=1.5.

Substitute x=1.5 in the given equation.

Therefore the maximum value of the function is 0.168 at x=1.5.

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step-by-step explanation: