If the function f(x) = left[ frac{(x - 2)^3}{ | vedclass

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(B) For the function $f(x) = \left[ \frac{(x - 2)^3}{a} \right] \sin(x - 2) + a \cos(x - 2)$ to be continuous on $[4, 6]$,the term inside the greatest

Vedclass · Accepted Answer

$65$

Vedclass · Answer

(B) For the function $f(x) = \left[ \frac{(x - 2)^3}{a} \right] \sin(x - 2) + a \cos(x - 2)$ to be continuous on $[4, 6]$,the term inside the greatest integer function must be constant,or the jump discontinuity must be nullified by the multiplying term $\sin(x - 2)$.In the interval $x \in [4, 6]$,the expression $(x - 2)^3$ ranges from $(4 - 2)^3 = 8$ to $(6 - 2)^3 = 64$.If $0 This occurs when $\frac{64}{a} \le 1$,which implies $a \ge 64$.If $a > 64$,then $\left[ \frac{(x - 2)^3}{a} \right] = 0$ for all $x \in [4, 6]$,making $f(x) = a \cos(x - 2)$,which is continuous.However,if $a = 64$,then at $x = 6$,$\left[ \frac{(6 - 2)^3}{64} \right] = [1] = 1$,and for $x Thus,the condition for continuity is $a > 64$. Among the given options,$65$ is the only value satisfying $a > 64$.

If the function $f(x) = \left[ \frac{(x - 2)^3}{a} \right] \sin(x - 2) + a \cos(x - 2)$ is continuous in $[4, 6]$,then the value of $a$ is (where $[.]$ denotes the greatest integer function).

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