The graph of f(x) = x^3 - x^2 - 6x is shown in the figure below.
How to plot graphs of cubic function?
- Calculate the x- and y-intercepts of the equation.
- To discover the local minima and maxima, take the derivative and set it to zero.
- To locate the point of inflection, use the second derivative.
Finding zeroes of the function:
f(x) = x^3 - x^2 - 6x
f(x) = x(x^2 - x - 6)
f(x) = x( x^2 - 3x + 2x - 6)
f(x) = x(x(x - 3) + 2(x - 3))
f(x) = x((x-3)(x+2))
From here we can say that f(x) will be zero at three points i.e. x = 0, x = 3, x = 2.
Finding local maxima and minima:
f'(x) = 3x^2 - 2x - 6
f'(x) = 0
x = 1.78, - 1.11
f(1.78) = -8.20
f(-1.11) = 4.06
The point of local maxima (-1.11 , 4.06)
The point of local minima (1.78 , -8.20)
Finding the point of inflexion:
f''(x) = 6x - 2
f''(x) = 0
x = 1/3
f(1/3) = -2.05
Point of inflexion = (0.33 , -2.05)
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