need this for a problem....
- Thread starter MBH_14
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What sort of polynomial is the derivative of a cubic?
If the cubic has max/min points at x = 2 and x = 5, then what does the derivative have at x = 2 and x = 5?
Once you've found a polynomial that fits the required form for the derivative, what do you get when you integrate (not forgetting the constant of integration)?
What adjustments do you need to make to this integrated result to make sure it passes through (x, y) = (2, 8) and (x, y) = (5, 1)?
Eliz.
If the cubic has max/min points at x = 2 and x = 5, then what does the derivative have at x = 2 and x = 5?
Once you've found a polynomial that fits the required form for the derivative, what do you get when you integrate (not forgetting the constant of integration)?
What adjustments do you need to make to this integrated result to make sure it passes through (x, y) = (2, 8) and (x, y) = (5, 1)?
Eliz.
Hello, MBH_14!
The point (2,8) is on the cubic; that is: \(\displaystyle f(2)\,=\,8\)
. . \(\displaystyle f(2)\:=\:a\cdot2^3\,+\,b\cdot2^2\,+\,c\cdot2\,+\,d\:=\:8\qquad\Rightarrow\qquad 8a\,+\,4b\,+\,2c\,+\,d\:=\:8\) .[1]
The point (5,1) is on the cubic; that is, \(\displaystyle f(5)\,=\,1\)
. . \(\displaystyle f(5)\:=\:a\cdot5^3\,+\,b\cdot5^2\,+\,c\cdot5\,+\,d\:=\:1\qquad\Rightarrow\qquad125a\,+\,25b\,+\,5c\,+\,d\:=\:1\) .[2]
There are extremes at (2,8) and (5,1).
. . Hence, when \(\displaystyle x = 2\) and \(\displaystyle x = 5\), the derivative is zero.
. . That is: \(\displaystyle f'(2)\,=\,0\) and \(\displaystyle f'(5)\,=\,0\)
The derivative is: .\(\displaystyle f'(x)\:=\:3ax^2 + 2bx + c\)
. . At \(\displaystyle x = 2:\;\;f'(2)\:=\:3a\cdot2^2\,+\,2b\cdot2\,+\,c\:=\:0\qquad\Rightarrow\qquad 12a\,+\,4b\,+\,c\:=\:0\) . [3]
. . At \(\displaystyle x = 5:\;\;f'(5)\:=\:3a\cdot5^2\,+\,2b\cdot6\,+\,c\:=\:0\qquad\Rightarrow\qquad 75a\,+\,10b\,+\,c\:=\:0\) . [4]
Solve the system of equations for \(\displaystyle a,\,b,\,c,\,d.\)
Let the cubic be: .\(\displaystyle f(x)\:=\:ax^3\,+\,bx^2\,+\,cx\,+\, d\)
The point (2,8) is on the cubic; that is: \(\displaystyle f(2)\,=\,8\)
. . \(\displaystyle f(2)\:=\:a\cdot2^3\,+\,b\cdot2^2\,+\,c\cdot2\,+\,d\:=\:8\qquad\Rightarrow\qquad 8a\,+\,4b\,+\,2c\,+\,d\:=\:8\) .[1]
The point (5,1) is on the cubic; that is, \(\displaystyle f(5)\,=\,1\)
. . \(\displaystyle f(5)\:=\:a\cdot5^3\,+\,b\cdot5^2\,+\,c\cdot5\,+\,d\:=\:1\qquad\Rightarrow\qquad125a\,+\,25b\,+\,5c\,+\,d\:=\:1\) .[2]
There are extremes at (2,8) and (5,1).
. . Hence, when \(\displaystyle x = 2\) and \(\displaystyle x = 5\), the derivative is zero.
. . That is: \(\displaystyle f'(2)\,=\,0\) and \(\displaystyle f'(5)\,=\,0\)
The derivative is: .\(\displaystyle f'(x)\:=\:3ax^2 + 2bx + c\)
. . At \(\displaystyle x = 2:\;\;f'(2)\:=\:3a\cdot2^2\,+\,2b\cdot2\,+\,c\:=\:0\qquad\Rightarrow\qquad 12a\,+\,4b\,+\,c\:=\:0\) . [3]
. . At \(\displaystyle x = 5:\;\;f'(5)\:=\:3a\cdot5^2\,+\,2b\cdot6\,+\,c\:=\:0\qquad\Rightarrow\qquad 75a\,+\,10b\,+\,c\:=\:0\) . [4]
Solve the system of equations for \(\displaystyle a,\,b,\,c,\,d.\)