## Applications of Derivatives: Tutorial: IITJEE Maths: Part II

Another set of “easy to moderately difficult” questions:

1. The function $y = \frac{}x{1+x^{2}}$ decreases in the interval (a) $(-1,1)$ (b) $[1, \infty)$ (c) $(-\infty, -1]$ (d) $(-\infty, \infty)$. There are more than one correct choices. Which are those?
2. The function $f(x) = \arctan (x) - x$ decreases in the interval (a) $(1,\infty)$ (b) $(-1, \infty)$ (c) $(-\infty, -\infty)$ (d) $(0, \infty)$. There is more than one correct choice. Which are those?
3. For $x>1$, $y = \log(x)$ satisfies the inequality: (a) $x-1>y$ (b) $x^{2}-1>y$ (c) $y>x-1$ (d) $\frac{x-1}{x}. There is more than one correct choice. Which are those?
4. Suppose $f^{'}(x)$ exists for each x and $h(x) = f(x) - (f(x))^{2} + (f(x))^{3}$ for every real number x. Then, (a) h is increasing whenever f is increasing (b) h is increasing whenever f is decreasing (c) h is decreasing whenever f is decreasing (d) nothing can be said in general. Find the correct choice(s).
5. If $f(x)=3x^{2}+12x-1$, when $-1 \leq x \leq 2$, and $f(x)=37-x$, when $2. Then, (a) $f(x)$ is increasing on $[-1,2]$ (b) $f(x)$ is continuous on $[-1,3]$ (c) $f^{'}(2)$ doesn’t exist (d) $f(x)$ has the maximum value at $x=2$. Find all the correct choice(s).
6. In which interval does the function $y=\frac{x}{\log(x)}$ increase?
7. Which is the larger of the functions $\sin(x) + \tan(x)$ and $f(x)=2x$ in the interval $(0?
8. Find the set of all x for which $\log {(1+x)} \leq x$.
9. Let $f(x) = |x-1| + a$, if $x \leq 1$; and, $f(x)=2x+3$, if $x>1$. If $f(x)$ has local minimum at $x=1$, then $a \leq$ ?
10. There are exactly two distinct linear functions (find them), such that they map $[-1,1]$ and $[0,2]$.

more later, cheers,

Nalin Pithwa.

This site uses Akismet to reduce spam. Learn how your comment data is processed.