The function $y = x^{2} + a x + b$ has a minimum at $x = 3$ and minimum value is $5$ . Find $a + b$ .

- 6

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14

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20

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8

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Solution

The correct option is D $8$
We have, $y = x^{2} + a x + b$
$\Rightarrow \frac{d y}{d x} = 2 x + a$
Given, $\frac{d y}{d x} = 0$ at $x = 3$
Therefore, $2 x + a = 0$
$\Rightarrow a = - 2 x = - 6$
Also, at $x = 3$ and $y = 5$
Therefore, $9 - 6 (3) + b = 5$
$b = 5 - 9 + 18 = 14$
Hence, $a + b = - 6 + 14 = 8$

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