Let a function f(x) defined on [3,6] be given by `f(x)={{:(,log_(e)[x],3 le x lt5),(,|log_(e)x|,5 le x lt 6):}` then f(x) is

Options

A. continuous and differentiable on [3,6]
B. continuous on [3,6] but not differentiable at x=4,5
C. differentiable on [3,6] but not continuous at x=4,5
D. none of these

Answer

D

Explanation

We have
`f(x)={{:(,log_(e)3,3 le x lt 4),(,log_(e)4,4 le x lt 5),(,log_(e)x,5 le x lt 6):}`
Clearly, f(x) is continuous and differentiable `[3,4) uu (4,5) uu (5,6)`
At x=4, we have
`underset(x to 4^(-))lim f(x)=log_(e) 3 and underset(x to 4^(+))lim f(x)=log_(e)4`
`therefore underset(x to 4^(-))lim f(x)ne underset(x to 4^(+))lim f(x)`
Thus, f(x) is neither continuous nor differentiable at x=4, At x=5, we have
So, f(x) is neither continuous nor differentiable at x=5.