Cubic function has 3 zeros
Q01. Diagram
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Q02. Cubic function has 3 zeros
Find y-intercept
- Let the expression be y = a*x^3 + b*x^2 + c*x + d
- Let a = 1.
- Let roots be r1 = 1, r2 = 2 and r3 = 3 as shown in diagram
- Method 1 : Use equation thoery
- Let roots be r1 = 1, r2 = 2 and r3 = 3 as shown in diagram
- b = -(r1 + r2 + r3) = -(1 + 2 + 3) = -6
- c = +(r1*r2 + r1*r3 + r2*r3) = +(1*2 + 1*3 + 2*3) = 11
- d = -(r1*r2*r3) = -(1*2*3) = -6
- Hence the expression is y = x^3 - 6*x^2 + 11*x - 6
- Method 2 : Use factors
- y = (x - 1)*(x - 2)*(x - 3)
- y = (x^2 - 3*x + 2)*(x - 3)
- y = x^3 - 3*x^2 + 2*x - 3*x^2 + 9*x - 6
- y = x^3 - 6*x^2 + 11*x - 6
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Q03. Find y'
First derivative y'
- y' = 3*x^2 - 12*x + 11
- y' = 0
- x1 = (12 - 4*Sqr(144 - 132))/2 = (12 - Sqr(12))/6 : Maximum
- x2 = (12 - 4*Sqr(144 - 132))/2 = (12 + Sqr(12))/6 : Minimum
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Q04. Find y"
Second derivative y"
- y" = 6*x - 12
- When x < 2, y" is negative and curve concave downward
- When x = 2, y" = 0 : point of inflection
- When x > 2, y" is positive and curve concave upward
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Q05. Discussion
Graphic solution
- The graphic solution is clear determine the signs of y, y' and y"
- The maximum and minimum points can only be estimated
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