Find the values of k so that the f is continuous the indicated point-f-x- - begin-cases- kx-2- if x le 2 3- if x - 2 end-cases- x-2

The given function is f-x-kx2-ifx-x2264-23-ifx-gt-2-xA0-The given function f is continuous at x-2limx-x2192-2f-x-limx-x2192-2f-x-f-2-x21D2-limx-x2192-2-kx2-limx-x2192-2-3-4k-x21D2-k-x22C5-22-3-4k-x21D2-4k-3-4k-x21D2-4k-3-x21D2-k-34Therefore- the required value of k is-xA0-34-

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Standard XII

Mathematics

RS Agarwal

Question

Find the values of $k$ so that the function $f$ is continuous at the indicated point:
$f (x) =$ ${\begin{matrix} k x^{2}, i f x \leq 2 3, i f x > 2 \end{matrix}$ at $x = 2$

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Solution

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The given function is $f (x) = {\begin{matrix} k x^{2}, if x \leq 2 3, if x > 2 \end{matrix}$
The given function $f$ is continuous at $x = 2$
$lim x \to 2 f (x) = lim x \to 2 f (x) = f (2)$
$\Rightarrow lim x \to 2 (k x^{2}) = lim x \to 2 (3) = 4 k$
$\Rightarrow k \cdot 2^{2} = 3 = 4 k$
$\Rightarrow 4 k = 3 = 4 k$
$\Rightarrow 4 k = 3$
$\Rightarrow k = \frac{3}{4}$
Therefore, the required value of $k$ is $\frac{3}{4}$ .

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