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If F(X) = X3 + Ax2 + Bx + C Has a Maximum at X = − 1 and Minimum at X = 3. Determine A, B and C ? - CBSE (Science) Class 12 - Mathematics

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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Solution If F(X) = X3 + Ax2 + Bx + C Has a Maximum at X = − 1 and Minimum at X = 3. Determine A, B and C ? Concept: Graph of Maxima and Minima.
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