Assertion (A): The minimum value of the expression $x^{2} + 2 b x + c$ is $c - b^{2}$ .
Reason (R): The first order derivative of the expression at $x = - b$ is zero.

Both A and R are true and R is the correct explanation of A

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Both A and R are true and R is not the correct explanation of A

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A is true but R is false

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A is false but R is true

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Solution

The correct option is A Both A and R are true and R is the correct explanation of A
$f (x) = x^{2} + 2 b x + c$
$f^{'} (x) = 2 x + 2 b$
For maxima or minima,
$f^{'} (x) = 0$
$\Rightarrow x = - b$
$f^{''} (x) = 2 > 0$
Hence f(x) has a minimum at $x = - b$
$f (- b) = c - b^{2}$

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Assertion (A): The minimum value of the expression x2+2bx+c is c−b2.Reason (R): The first order derivative of the expression at x=−b is zero.

The correct option is A Both A and R are true and R is the correct explanation of Af(x)=x2+2bx+cf′(x)=2x+2bFor maxima or minima,f′(x)=0⇒x=−bf′′(x)=2>0Hence f(x) has a minimum at x=−bf(−b)=c−b2

Assertion (A): The minimum value of the expression $x^{2} + 2 b x + c$ is $c - b^{2}$ .
Reason (R): The first order derivative of the expression at $x = - b$ is zero.

The correct option is A Both A and R are true and R is the correct explanation of A
$f (x) = x^{2} + 2 b x + c$
$f^{'} (x) = 2 x + 2 b$
For maxima or minima,
$f^{'} (x) = 0$
$\Rightarrow x = - b$
$f^{''} (x) = 2 > 0$
Hence f(x) has a minimum at $x = - b$
$f (- b) = c - b^{2}$