If f x - eft begin matrix 3-x-k- - text for x k a - 2-2- frac sin x - k x - k - - text for x - kend matrix right has minimum at x - k- thenA- - a -2D- 12

The correct option is C |a| gt; 2 lim_x → kf(x)=3+h⇒ f(k)=3 f(k^ ) gt; f(k) ........always true as f(k^ ) = 3 + |x h| and f(k^+) gt; f(k) ⇒ a^2 2+ ...

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Byju's Answer

Standard XII

Mathematics

Convexity

If f x =⟨ eft...

Question

If $f (x) = {\begin{matrix} 3 + | x - k |, & for x \leq k a^{2} - 2 + \frac{s i n (x - k)}{x - k}, & for x > k \end{matrix}$ has minimum at x = k, then

$a \in R$

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$| a | < 2$

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$| a | > 2$

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$1 < | a | > 2$

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Solution

The correct option is C
$| a | > 2$

${lim}_{x \to k} f (x) = 3 + h \Rightarrow f (k) = 3$
$f (k^{-}) > f (k) . . . . . . . . always true as f (k^{-}) = 3 + | x - h |$

and $f (k^{+}) > f (k)$
$\Rightarrow a^{2} - 2 + 1 > 3$
$| a | > 2$

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If f(x)={3+|x−k|,for x≤ka2−2+sin(x−k)x−k,for x>k has minimum at x = k, then

The correct option is C |a|>2 limx→kf(x)=3+h⇒f(k)=3 f(k−)>f(k) ........always true as f(k−)=3+|x−h| and f(k+)>f(k) ⇒a2−2+1>3 |a|>2

If $f (x) = {\begin{matrix} 3 + | x - k |, & for x \leq k a^{2} - 2 + \frac{s i n (x - k)}{x - k}, & for x > k \end{matrix}$ has minimum at x = k, then

The correct option is C
$| a | > 2$

${lim}_{x \to k} f (x) = 3 + h \Rightarrow f (k) = 3$
$f (k^{-}) > f (k) . . . . . . . . always true as f (k^{-}) = 3 + | x - h |$

and $f (k^{+}) > f (k)$
$\Rightarrow a^{2} - 2 + 1 > 3$
$| a | > 2$