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secondary 4 | A Maths
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y = 2x^3 -9x^2+5
The gradient function of the curve is given by
dy/dx = 6x2 - 18x
The gradient itself is an expression in x; the minimum value of the gradient, or the minimum value of this expression 6x2 - 18x, is found by differentiating this expression one more times and setting the new expression is zero
[For a minimum value of something, its derivative is equal to zero]
d2y/dx2 = 12x - 18
So,
12x - 18 = 0
12x = 18
x = 3/2
Minimum gradient
= 6 (3/2)^2 - 18 (3/2)
= 27/2 - 27
= -27/2
= -13 1/2
Or you could try d3y/dx3 = 12 and this indicates that the particular value of dy/dx must certainly be a minimum.
Alternatively, you could complete the square on dy/dx
dy/dx
= 6x2 - 18x
= 6 (x2 - 3x)
= 6 [x2 - 3x + 1.5^2 - 1.5^2]
= 6 [(x - 1.5)^2 - 2.25]
= 6 (x - 1.5)^2 - 13.5
And you can see that this “quadratic” expression is going to have a clear minimum of -13.5 when x = 1.5.
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