If fx-3-x-k- x- k a2-2-sinx-kx-k x- k - hasminimum at x - k- then

The correct option is C |a| > 2 f(k) = 3 f(k+h) =lim_h→ 0 ( a^2 2 +sin hh )⇒lim_h→ 0f(k+h) = a^2 1 lim_h→ 0f(k h) =lim_h→ 0(3 + |k h k| ...

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Byju's Answer

Standard XII

Mathematics

Convexity

If fx=3+|x-...

Question

If $f (x) = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} 3 + | x - k |, & x \leq k a^{2} - 2 + \frac{sin (x - k)}{x - k} & x > k \end{matrix}$ has
minimum at $x = k$ , then

a ε 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 $| a | > 2$
$f (k) = 3$
$f (k + h) = {lim}_{h \to 0} (a^{2} - 2 + \frac{s i n h}{h}) \Rightarrow {lim}_{h \to 0} f (k + h) = a^{2} - 1$
${lim}_{h \to 0} f (k h) = {lim}_{h \to 0} (3 + | k - h - k |) = {lim}_{h \to 0} (3 + | - h |) = 3$
$\Rightarrow a^{2} - 1 > 3$
$\Rightarrow a^{2} > 4 \Rightarrow | a | > 2$
Hence, option 'C' is correct.

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Similar questions

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

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Find the value of k if x-3 is a factor of k²x³-kx²+3kx-k.

Q. If

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is a monotonically decreasing function of x in the largest possible interval (-2, -2/3). Then .

Find the value of k, if x - 1 is a factor p(x) in each of the following cases:
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If f(x)=⎧⎪⎨⎪⎩3+|x−k|,x≤ka2−2+sin(x−k)x−kx>k hasminimum at x=k, then

The correct option is C |a|>2f(k)=3f(k+h)=limh→0(a2−2+sinhh)⇒limh→0f(k+h)=a2−1limh→0f(kh)=limh→0(3+|k−h−k|)=limh→0(3+|−h|)=3⇒a2−1>3⇒a2>4⇒|a|>2Hence, option 'C' is correct.

If $f (x) = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} 3 + | x - k |, & x \leq k a^{2} - 2 + \frac{sin (x - k)}{x - k} & x > k \end{matrix}$ has
minimum at $x = k$ , then