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First of all we want to think about the general behaviour of the function. Does it grow as x grows? Does it oscillate? Does it tend to a limit? We also need to think about if there are any worrying points...

If helpful draw a diagram to see what is going on, but keep the time in mind! From your sketch it should be apparent that c must be positive and that the single point of contact of the curves is above the...

A point a is a stationary point if and only if f'(a) = 0.
Let's compute the derivative:
f'(x) = (2x^3 − kx^2 + 2x − k)' = 6x^2 - 2kx + 2.  (-k is a constant so it cancells out)
f'(x) = 0 i...

The easiest way to do this is to notice that f(x-2) is just f(x) translated by two in the positive x-direction - it is not necessary to calculate what f(x-2) is at any point!
So we calculate f'(x) =...

Firstly, we want to notice that this question only involves powers of 2^x, ie.
(2^x)^3 + 4 = (2^x)^2 + 4(2^x)
so we can use the substitution u = 2^x to give
u^3 - u^2 - 4u + 4 = 0
The...