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A function f is defined by f(x)=x*e^(-2x) with domain 0 <= x <= 10

Find all values of x for which the raph of f is increasing and all values of x for which the graph is decreasing.

Give the x- and y- coordinates of all absolute maximum and minimum points on the graph of f and justify.

 

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Find all values of x for which the graph of f is increasing and all values of x for which the graph is decreasing.

Differentiate, then factorise. Set f'(x) > 0 and f'(x) < 0, respectively.
Note that e^(-2x) is always positive, so the other part is what determines the sign of f'(x).

Give the x- and y- coordinates of all absolute maximum and minimum points on the graph of f and justify.

Set f'(x) = 0.
Again, e^(-2x) cannot be zero, so the other part must be.
Test endpoints: f(0) and f(10).