Left Hand Derivative

Q. If $f\left(x\right)=\{\begin{array}{cc}{4}^x,& -1\le x<1\\ 5-x,& 1\le x\le 5\end{array},$ then

1. $f\left(x\right)$ is continuous but not differentiable at $x=1$
2. $f\left(x\right)$ is differentiable at $x=1$
3. $f\left(x\right)$ is discontinuous at $x=1$
4. None of these

Q. The left-hand derivative of $f\left(x\right)=\left[x\right]sin\left(\pi x\right)$ at x= k, k is an integer and [x] = greatest integer, is

1. $(-1{)}^k(k-1)\pi$
2. $(-1{)}^{k-1}(k-1)\pi$
3. $(-1{)}^{k}k\pi$
4. $(-1{)}^{k-1}k\pi$

Q. If the function $f\left(x\right)=\{\begin{array}{ccc}2a^{2}x-1& ;& x\le 1\\ x^2+x+b& ;& x>1\end{array}$
is differentiable everywhere, where $a,b\in R$, then the value of $2a^2+b$ is

Q. The left-hand derivative of f(x) = [x] sin ($\pi$ x) at x = k, k is an integer and [x] = greatest integer $\le$ x, is

[IIT Screening 2001]

1. $(-1{)}^k(k-1)\pi$
2. $(-1{)}^{k-1}(k-1)\pi$
3. $(-1{)}^{k}k\pi$
4. $(-1{)}^{k-1}k\pi$