My name is Barnum and I am a second year maths undergraduate at Cambridge University. I have A-Levels in Maths, Further Maths, Physics, Chemistry and German, all at grade A*.
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This is a typical A/S level question, which can be broken down into three key steps.
1) Differentiation: Remember that at a stationary point, the gradient of the curve is zero. Therefore, we first have to find the gradient, which means differentiating. (Multiply the coefficient of x by the power, and then reduce the power by 1)
y = 2x3 + 3x2 - 12x + 1
dy/dx = 6x2 + 6x - 12
2) Solve Quadratic: To find where the gradient is zero (dy/dx = 0), we must solve the quadratic equation,
dy/dx = 6x2 + 6x - 12 = 0.
This quadratic can be most easily solved by factorisation (alternative methods are completing the square and using the quadratic formula).
Firstly we can take 6 out as a common factor, which results in 6(x2 + x - 2) = 0. We can then factorise the terms in brackets which results in 6(x+2)(x-1) = 0.
This gives the two solutions x = -2, x = 1.
3) Substitution: So far, we have only found the x coordinate of the stationary point, but to answer the question fully we need the y coordinate as well. We do this by substituting x = -2, and x = 1 into the original equation of the curve, y = 2x3 + 3x2 - 12x + 1
When x = -2, we can calcuclate that y = 21. Similarly when x = 1, y = -6.
Thus the coordinates of the stationary points are (-2, 21) and (1, -6).see more