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Firstly, to find the stationary points of a curve you must differentiate the equation of the curve. To do this each x component is multiplied by its current power and then the power is decreased by one. A...

∫(cos² 2x *sin³ 2x)dx u = cos2x - u =(du/dx) = -2sin2x - differentiate u dx = du/(-2sin(2x)) - dx = -1/2 ∫cos²2x * sin²2x du - sub in dx-1/2 ∫u²(1-u<...

This equation has no real roots, as when we test the equation with the formula b² - 4ac, we get 2² -415 = 4 - 20 = -16. As this is negative we can be assured that this equat...

y = 4x³ - 7x -10The gradient of the function at any point can be found using its derivative:dy/dx = 12x² - 7The gradient of the function, m₁, at (2,8) is equal to the grad...

f(x) = 4x² - 16ln(x-1) -10, f'(x) = 8x - 16/(x-1), so if f'(x)=0, then 8x - 16/(x -1)=0, 8x(x-1) - 16 = 0, 8x²- 8x - 16 =0, 8(x²- x - 2) = 0, x² - x - 2 = 0, (x...