MCQ
The function $f(x) = ({x^2} - 1)|{x^2} - 3x + 2| + \cos (|x|)$ is not differentiable at
- A$-1$
- B$0$
- C$1$
- ✓$2$
$\therefore \,|{x^2} - 3x + 2| = |(x - 1)(x - 2)|$
Hence is not differentiable at $x = 1$ and $2$
Now $f(x) = ({x^2} - 1)|{x^2} - 3x + 2|\cos (|x|)$ is not differentiable at $x = 2$
For $1 < x < 2$, $f(x) = - ({x^2} - 1)({x^2} - 3x + 2) + \cos x$
For $2 < x < 3$, $f(x) = + ({x^2} - 1)({x^2} - 3x + 2) + \cos x$
$Lf'(x) = - ({x^2} - 1)(2x - 3) - 2x({x^2} - 3x + 2) - \sin x$
$Lf'(2) = - 3 - \sin 2$
$Rf'(x) = ({x^2} - 1)(2x - 3) + 2x({x^2} - 3x + 2) - \sin x$
$Rf'(2) = (4 - 1)(4 - 3) + 0 - \sin 2 = 3 - \sin 2$
Hence $Lf'(2) \ne Rf'(2)$.
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$\lim _{n \rightarrow 0^{+}} \int_n^{1-n} t^{-3}(1-t)^{a-1} d t$
exists. Let this limit be $g(a)$. In addition, it is given that the function $g(a)$ is differentiable on $(0,1)$.
$1.$ The value of $g\left(\frac{1}{2}\right)$ is
$(A)$ $\pi$ $(B)$ $2 \pi$ $(C)$ $\frac{\pi}{2}$ $(D)$ $\frac{\pi}{4}$
$2.$ The value of $g ^{\prime}\left(\frac{1}{2}\right)$ is
$(A)$ $\frac{\pi}{2}$ $(B)$ $\pi$ $(C)$ $-\frac{\pi}{2}$ $(D)$ $0$
Give the answer question $1$ and $2.$
$(A)$ $P\left[X_1^c \mid x\right]=\frac{3}{16}$
$(B)$ $P [$ Exactly two engines of the ship are functioning $\mid X ]=\frac{7}{8}$
$(C)$ $P\left[X \mid X_2\right]=\frac{5}{16}$
$(D)$ $P\left[X \mid X_1\right]=\frac{7}{16}$