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Question
If f(x) = x3 + ax2 + bx + c has a maximum at x = \[-\] 1 and minimum at x = 3. Determine a, b and c ?
Solution
\[\text { We have,} \]
\[f\left( x \right) = x^3 + a x^2 + bx + c\]
\[ \Rightarrow f'\left( x \right) = 3 x^2 + 2ax + b\]
\[\text { As,} f\left( x \right) \text { is maximum at x = - 1 and minimum at x = 3 }. \]
\[\text { So,} f\left( - 1 \right) = 0 \text { and } f\left( 3 \right) = 0\]
\[ \Rightarrow 3 \left( - 1 \right)^2 + 2a\left( - 1 \right) + b = 0\text { and }3 \left( 3 \right)^2 + 2a\left( 3 \right) + b = 0\]
\[ \Rightarrow 3 - 2a + b = 0 . . . . . \left( i \right)\]
\[\text { and }27 + 6a + b = 0 . . . . . \left( ii \right)\]
\[\left( ii \right) - \left( i \right), \text { we get }\]
\[27 - 3 + 6a + 2a = 0\]
\[ \Rightarrow 8a = - 24\]
\[ \Rightarrow a = - 3\]
\[\text { Substituting a } = - 3 \text { in } \left( i \right), \text { we get }\]
\[3 - 2\left( - 3 \right) + b = 0\]
\[ \Rightarrow 3 + 6 + b = 0\]
\[ \Rightarrow b = - 9\]
\[\text { And }, c \in R\]
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