JEE MAIN - Mathematics (2021 - 27th July Evening Shift - No. 12)
Let $$f:[0,\infty ) \to [0,3]$$ be a function defined by
$$f(x) = \left\{ {\matrix{ {\max \{ \sin t:0 \le t \le x\} ,} & {0 \le x \le \pi } \cr {2 + \cos x,} & {x > \pi } \cr } } \right.$$
Then which of the following is true?
$$f(x) = \left\{ {\matrix{ {\max \{ \sin t:0 \le t \le x\} ,} & {0 \le x \le \pi } \cr {2 + \cos x,} & {x > \pi } \cr } } \right.$$
Then which of the following is true?
f is continuous everywhere but not differentiable exactly at one point in (0, $$\infty$$)
f is differentiable everywhere in (0, $$\infty$$)
f is not continuous exactly at two points in (0, $$\infty$$)
f is continuous everywhere but not differentiable exactly at two points in (0, $$\infty$$)
Explanation
Graph of $$\max \{ \sin t:0 \le t \le x\} $$ in $$x \in [0,\pi ]$$
_27th_July_Evening_Shift_en_12_1.png)
& graph of cos x for $$x \in [\pi ,\infty )$$
_27th_July_Evening_Shift_en_12_2.png)
So graph of
$$f(x) = \left\{ {\matrix{ {\max \{ \sin t:0 \le t \le x\} ,} & {0 \le x \le \pi } \cr {2 + \cos x,} & {x > \pi} \cr } } \right.$$
_27th_July_Evening_Shift_en_12_3.png)
f(x) is differentiable everywhere in (0, $$\infty$$)
& graph of cos x for $$x \in [\pi ,\infty )$$
So graph of
$$f(x) = \left\{ {\matrix{ {\max \{ \sin t:0 \le t \le x\} ,} & {0 \le x \le \pi } \cr {2 + \cos x,} & {x > \pi} \cr } } \right.$$
f(x) is differentiable everywhere in (0, $$\infty$$)
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