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.

check above link.

graphically?   well, you graph them both, the solutions is where they both intersect each other, since there are two solutions, meaning they will meet twice, or namely the linear will run into the quadratic curve twice, and where they meet, those x,y coordinates are the solutions to the system.