The least integral value of k for which fx - e-x-k-8x2 is monotonically decreasing for all x- is

F(x) = e^ x√(k+8x^2) f'(x) = √(k+8x^2)·( e^ x) e^ x·16x2 √(k+8x^2)(k + 8x^2) ⇒ f'(x) = (k+8x^2)( e^ x) 8xe^ x(k+8x^2)^3/2 ⇒ f'(x) = e^ x(8x^2 + 8x +k)(k+8x^ ...

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Standard XII

Mathematics

Rolle's Theorem

The least int...

Question

The least integral value of $k$ for which $f (x) = \frac{e^{- x}}{\sqrt{k + 8 x^{2}}}$ is monotonically decreasing for all $x \in R,$ is

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Solution

The correct option is B $2$
$f (x) = \frac{e^{- x}}{\sqrt{k + 8 x^{2}}}$
$f^{'} (x) = \frac{\sqrt{k + 8 x^{2}} \cdot (- e^{- x}) - e^{- x} \cdot \frac{16 x}{2 \sqrt{k + 8 x^{2}}}}{(k + 8 x^{2})}$
$\Rightarrow f^{'} (x) = \frac{(k + 8 x^{2}) (- e^{- x}) - 8 x e^{- x}}{(k + 8 x^{2})^{3 / 2}}$
$\Rightarrow f^{'} (x) = \frac{- e^{- x} (8 x^{2} + 8 x + k)}{(k + 8 x^{2})^{3 / 2}}$
Since $f (x)$ is monotonically decreasing for all $x \in R,$
$∴ f^{'} (x) \leq 0$ for all $x \in R$
$\Rightarrow \frac{- e^{- x} (8 x^{2} + 8 x + k)}{(k + 8 x^{2})^{3 / 2}} \leq 0$
$\Rightarrow 8 x^{2} + 8 x + k \geq 0 \forall x \in R$
$\Rightarrow D \leq 0$
$\Rightarrow 64 - 32 k \leq 0$
$\Rightarrow k \geq 2$
Least integral value of $k$ is $2.$

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The least integral value of k for which f(x)=e−x√k+8x2 is monotonically decreasing for all x∈R, is

The least integral value of $k$ for which $f (x) = \frac{e^{- x}}{\sqrt{k + 8 x^{2}}}$ is monotonically decreasing for all $x \in R,$ is