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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 = 0The...
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