MCQ
If $f(x) = k{x^3} - 9{x^2} + 9x + 3$ is monotonically increasing in each interval, then
- A$k < 3$
- B$k \le 3$
- ✓$k > 3$
- DNone of these
$\therefore \Delta = {b^2} - 4ac < 0\,\,\,,k > 0$
$i.e.,\;\;36 - 12k < 0$ or $k > 3$.
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$f(x)=\left\{\begin{array}{cc}x^2 \sin \left(\frac{\pi}{x^2}\right) & \text { if } x \neq 0 \\ 0, & \text { if } x=0\end{array}\right.$
Then which of the following statements is $TRUE$?
$(A)$ $f^{\prime \prime}(x)$ exists for all $x \in(0, \infty)$
$(B)$ $f^{\prime}(x)$ exists for all $x \in(0, \infty)$ and $f^{\prime}$ is continuous on $(0, \infty)$, but not differentiable on $(0, \infty)$
$(C)$ there exists $\alpha>1$ such that $\left|f^{\prime}(x)\right|<|f(x)|$ for all $x \in(\alpha, \infty)$
$(D)$ there exists $\beta>0$ such that $|f(x)|+\left|f^{\prime}(x)\right| \leq \beta$ for all $x \in(0, \infty)$