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Q.
Let $f(x)=\frac{2 x^3}{3}-(x \ln |x|-x)-k x+10$. If $f(x)$ is strictly increasing for every $x \in R-\{0\}$, then the maximum integral value of $k$ is
Application of Derivatives
Solution:
$f ^{\prime}( x )=2 x ^2-\ln | x |- k \geq 0 \forall x \in R -\{0\}$
$k \leq 2 x ^2-\ln | x | \forall x \in R -\{0\}$
Hence, $k \leq$ minimum value of $2 x ^2-\ln | x |$
$\therefore k \leq \frac{1}{2}+\ln 2$
Hence, maximum integral value of $k$ is 1 .