The minimum value of x2-11-x2 occurs at

Let f(x)= nbsp;x^2+11+x^2 nbsp; On differentiating w.r.t. x, we get f'(x)=2x 1(1+x^2)^2· 2x For a minimum, put f'(x)=0 ⇒ 2x( (1+x^2)^2 1(1+x^2)^2)=0 ⇒ x=0 ...

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Standard XII

Mathematics

Chain Rule of Differentiation

The minimum v...

Question

The minimum value of $x^{2} + \frac{1}{1 + x^{2}}$ occurs at

x = 4

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x = 3

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x = 0

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x = 1

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Solution

The correct option is C $x = 0$
Let $f (x) = x^{2} + \frac{1}{1 + x^{2}}$
On differentiating w.r.t. $x$ , we get $f^{'} (x) = 2 x - \frac{1}{(1 + x^{2})^{2}} \cdot 2 x$
For a minimum, put $f^{'} (x) = 0$
$\Rightarrow 2 x (\frac{(1 + x^{2})^{2} - 1}{(1 + x^{2})^{2}}) = 0$
$\Rightarrow x = 0$

So, the function has the minimum value at $x = 0$

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Chain Rule of Differentiation

Standard XII Mathematics

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The minimum value of x2+11+x2 occurs at

The correct option is C x=0Let f(x)=x2+11+x2 On differentiating w.r.t. x, we get f′(x)=2x−1(1+x2)2⋅2x For a minimum, put f′(x)=0 ⇒2x((1+x2)2−1(1+x2)2)=0 ⇒x=0 So, the function has the minimum value at x=0

The minimum value of $x^{2} + \frac{1}{1 + x^{2}}$ occurs at