# find constants

#### euclid2

find constants a,b, and c such that the graph of $$\displaystyle f(x)=x^3+ax^2+bx+c$$ will increase to the point $$\displaystyle (-3,18)$$, decrease to the point (1,-14) and then continue increasing

find constants a,b, and c such that the graph of $$\displaystyle f(x)=x^3+ax^2+bx+c$$ will increase to the point $$\displaystyle (-3,18)$$, decrease to the point (1,-14) and then continue increasing
Find the derivative.

$$\displaystyle f'(x) = 3x^2 + 2ax + b$$

Set this equal to zero at the points $$\displaystyle x=-3$$ and $$\displaystyle x=1$$ which will be our critical points.

We get $$\displaystyle 27 - 6a + b = 0$$ and $$\displaystyle 3 + 2a + b = 0$$

Solving these simultaneous equations gives us...

$$\displaystyle a = 3$$ and $$\displaystyle b = -9$$.

You can check the second derivative to see that the points $$\displaystyle (-3,18)$$ and $$\displaystyle (1,-14)$$ are indeed a max and a min.

Hence we have $$\displaystyle f(x) = x^3 + 3x^2 - 9x + c$$.

Now just find a value of $$\displaystyle c$$ that solves...

$$\displaystyle -14 = 1 + 3 - 9 + c$$ (for the point $$\displaystyle (1,-14)$$...)

Graph of said function below after you find the $$\displaystyle c$$ value...

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