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If f(x) = {{:(-x-pi/2" , " x le - pi...

If `f(x) = {{:(-x-pi/2" , " x le - pi/2),(-cos x" , " -pi/2 lt x le 0",""then"),(x-1", "0lt x le 1),(ln x", "x gt 1):}`

A

f(x) is continuous at x = -`pi/2`

B

f(x) is not differentiable at x = 0

C

f(x) is differentiable at x = 1

D

f(x) is differentiable at x = - `3/2`

Text Solution

Verified by Experts

The correct Answer is:
A, B, C, D
`f(x) ={{:(-x-pi/2", " x le -pi/2),(-cosx", " -pi/2 lt x le 0),( x - 1", " 0lt xle 1),(log x", " x gt 1):}`
Continuity at ` x = - pi/2`,
`f(-pi/2) =- (-pi/2) = pi/2 = 0`
RHL ` = underset( h to 0) lim -cos (-pi/2+h) = 0`
`:. " Continuous at " x= (-pi/2)`.
Continuity at x = 0
f(0) =- 1
RHL ` =underset( h to 0) lim (0+h)-1=- 1`
`:.` Continuous at x = 0 .
Continuity at x = 1,
f (1) = 0
RHL ` = underset( h to 0) lim log (1+h) = 0`
`:.` Continuous at x = 1
`f'(x) = {{:(-1", " x le -pi/2),(sin x", "-pi/2 lt x le 0),(" 1, " 0 lt x le 1),(1/x", " x gt 1):}`
Differentiable at x = 0, LHD = 0, RHD = 1
`:.` Not differentiable at x = 0
Differentiable at x = 1, LHD = 1, RHD = 1
`:.` Differentiable at x = 1.
Also, for ` x =- 3/2`
`rArr " " f(x) =- x - pi/2`
` :. ` Differentiable at ` x = - 3/2 `
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