Let f:[0, ∞) arrow[0,3] be a function defined by f(x)= begincases max sin t: 0 ≤ t ≤ x 0 ≤ x ≤ π 2+ cos x, xπ endcases Then which of the following is true?

Question

Let f:[0, ∞) arrow[0,3] be a function defined by f(x)= begincases max sin t: 0 ≤ t ≤ x 0 ≤ x ≤ π 2+ cos x, x>π endcases Then which of the following is true? (A) f is continuous everywhere but not differentiable exactly at one point in (0, ∞) (B) f is differentiable everywhere in (0, ∞) (C) f is not continuous exactly at two points in (0, ∞) (D) f is continuous everywhere but not differentiableexactly at two points in (0, ∞)

Tardigrade Admin · Accepted Answer

Graph of max sin t: 0 ≤ t ≤ x in x ∈[0, π]

So graph of f(x)= begincases max sin t: 0 ≤ t ≤ x 0 ≤ x ≤ π 2+ cos x, x> h endcases

f is differentiable everywhere in (0, ∞)

Q. Let $f : [0, \infty) \to [0, 3]$ be a function defined by
$f (x) = {max {sin t : 0 \leq t \leq x}, 2 + cos x, 0 \leq x \leq π x > π$
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 differentiableexactly at two points in $(0, \infty)$

Graph of $max {sin t : 0 \leq t \leq x}$ in $x \in [0, π]$

So graph of
$f (x) = {max {sin t : 0 \leq t \leq x}, 2 + cos x, 0 \leq x \leq π x > h$

$f$ is differentiable everywhere in $(0, \infty)$

Q. Let f:[0,∞)→[0,3] be a function defined by f(x)={max{sint:0≤t≤x},2+cosx,​0≤x≤πx>π​ Then which of the following is true?

f is continuous everywhere but not differentiable exactly at one point in (0,∞)

f is differentiable everywhere in (0,∞)

f is not continuous exactly at two points in (0,∞)

f is continuous everywhere but not differentiableexactly at two points in (0,∞)