Find the values of constants \( a, b \), and \( c \) so that the graph of \( y=a x^{3}+b x^{2}+c x \) has a local maximum at \( x=-2 \), local minimum at \( x=-4 \), and inflection point at \( (-3,18) \). \[ a=\square=\square \quad c=\square \] (Simplify your answers. Type integers or simplified fractions.)
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So, we take the derivative of the function: \(y'=3ax^2+2bx+c\) Setting this equal to zero and solving for x gives us the x-values of the local maximum and minimum: \(0=3ax^2+2bx+c\) This is a quadratic equation, and we know that its solutions are -2 and -4. Show more…
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