Given the following properties of a function $f(x)$:
$(i)$ $f(x)$ is continuous and defined for all real numbers.
$(ii)$ $f'(-5) = 0$; $f'(2)$ is not defined and $f'(4) = 0$.
$(iii)$ $(-5, 12)$ is a point on the graph of $f(x)$.
$(iv)$ $f''(2)$ is undefined,but $f''(x)$ is negative everywhere else.
$(v)$ The signs of $f'(x)$ are given by the following number line:
$f'(x)$ is positive for $x < -5$,negative for $-5 < x < 2$,positive for $2 < x < 4$,and negative for $x > 4$.
On the possible graph of $y = f(x)$,we have:
$(i)$ $f(x)$ is continuous and defined for all real numbers.
$(ii)$ $f'(-5) = 0$; $f'(2)$ is not defined and $f'(4) = 0$.
$(iii)$ $(-5, 12)$ is a point on the graph of $f(x)$.
$(iv)$ $f''(2)$ is undefined,but $f''(x)$ is negative everywhere else.
$(v)$ The signs of $f'(x)$ are given by the following number line:
$f'(x)$ is positive for $x < -5$,negative for $-5 < x < 2$,positive for $2 < x < 4$,and negative for $x > 4$.
On the possible graph of $y = f(x)$,we have:
- A$x = -5$ is a point of relative minima.
- B$x = 2$ is a point of relative maxima.
- C$x = 4$ is a point of relative minima.
- DThe graph of $y = f(x)$ must have a geometrical sharp corner at $x = 2$.
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Which of the following is not true?
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View SolutionIn the following $[x]$ denotes the greatest integer less than or equal to $x$. Match the functions in Column $I$ with the properties in Column $II$.
Column $I$ Column $II$ $(A)$ $f(x) = x|x|$ $(p)$ continuous in $(-1, 1)$ $(B)$ $f(x) = \sqrt{|x|}$ $(q)$ differentiable in $(-1, 1)$ $(C)$ $f(x) = x + [x]$ $(r)$ strictly increasing in $(-1, 1)$ $(D)$ $f(x) = |x - 1| + |x + 1|$ $(s)$ not differentiable at least at one point in $(-1, 1)$
| Column $I$ | Column $II$ |
|---|---|
| $(A)$ $f(x) = x|x|$ | $(p)$ continuous in $(-1, 1)$ |
| $(B)$ $f(x) = \sqrt{|x|}$ | $(q)$ differentiable in $(-1, 1)$ |
| $(C)$ $f(x) = x + [x]$ | $(r)$ strictly increasing in $(-1, 1)$ |
| $(D)$ $f(x) = |x - 1| + |x + 1|$ | $(s)$ not differentiable at least at one point in $(-1, 1)$ |
If $f(x) = \begin{cases} x^{3}-3x+2, & x < 2 \\ x^{3}-6x^{2}+9x+2, & x \geq 2 \end{cases}$, then:
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View SolutionLet $f:R \to R$ be a continuously differentiable function such that $f(2) = 6$ and $f'(2) = \frac{1}{48}$. If $\int_6^{f(x)} 4t^3 \,dt = (x - 2)g(x)$,then $\lim_{x \to 2} g(x)$ is equal to
DifficultJEE MAIN 2019
View SolutionMatch the items of List-$I$ with those of List-$II$.
List-$I$ List-$II$ $A$. If $y = |x| + |x - 2|$, then at $x = 2$, $\frac{dy}{dx} =$ $I$. $2$ $B$. If $f(x) = |\cos 2x|$, then $f'(\frac{\pi}{4} +) =$ $II$. $0$ $C$. If $f(x) = \sin(\pi[x])$, where $[x]$ is the greatest integer function, then $f'(1-) =$ $III$. $-2$ $D$. If $f(x) = \log|x - 1|$, $x \neq 1$, then $f'(\frac{1}{2}) =$ $IV$. does not exist
| List-$I$ | List-$II$ |
| $A$. If $y = |x| + |x - 2|$, then at $x = 2$, $\frac{dy}{dx} =$ | $I$. $2$ |
| $B$. If $f(x) = |\cos 2x|$, then $f'(\frac{\pi}{4} +) =$ | $II$. $0$ |
| $C$. If $f(x) = \sin(\pi[x])$, where $[x]$ is the greatest integer function, then $f'(1-) =$ | $III$. $-2$ |
| $D$. If $f(x) = \log|x - 1|$, $x \neq 1$, then $f'(\frac{1}{2}) =$ | $IV$. does not exist |
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