MCQ
For two events $A$ and $B$,
$P(A)+P(B)-P(A$ and $B)=P(A)$, then __________ .
$P(A)+P(B)-P(A$ and $B)=P(A)$, then __________ .
- A$P(A \mid B)=0$
- ✓$P ( A \mid B )=1$
- C$P ( B \mid A )=0$
- D$P(B \mid A)=1$
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$f(x)=\left\{\begin{array}{rc}x^5+5 x^4+10 x^3+10 x^2+3 x+1, & x<0 \\ x^2-x+1, & 0 \leq x<1 \\ \frac{2}{3} x^3-4 x^2+7 x-\frac{8}{3}, & 1 \leq x<3 \\ (x-2) \log _e(x-2)-x+\frac{10}{3}, & x \geq 3\end{array}\right.$
Then which of the following options is/are correct?
$(1)$ $f^{\prime}$ has a local maximum at $x =1$ $(2)$ $f$ is onto
$(3)$ $f$ is increasing on $(-\infty, 0)$ $(4)$ $f^{\prime}$ is $NOT$ differentiable at $x =1$