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Which of the following statement is true...

Which of the following statement is true for the function `f(x)={{:(sqrt(x),","x ge 1),(x^(3) ,","0 le x lt 1),((x^(3))/(3)-4x,"," x lt 0):}`

A

It is monotonic increasing `AA x in R`

B

f'(x) fails to exist for 3 distinct real values of x

C

f'(x) changes its sign twice as x varies from `(-oo, oo)`

D

Function attains its extreme values at `x_(1) and x_(2)`, such that `x_(1)x_(2) gt 0`

Text Solution

Verified by Experts

The correct Answer is:
C
`f(x)={(sqrt(x),x ge 1),(x^(3),0 le x le 1),((x^(3))/(3)-4x,x lt 0):}`
`f.(x)={((1)/(2sqrt(x)), x ge 1), (3x^(2), 0 le x le 1),(x^(2)-4, x lt 0):}`
f.(x) changes its sign two times at `x=0, -2`
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