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

Let a function f(x) defined on [3,6] be given by `f(x)={{:(,log_(e)[x]

1.	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) isA. continuous and differentiable on [3,6]B. continuous on [3,6] but not differentiable at x=4,5C. differentiable on [3,6] but not continuous at x=4,5D. none of these
Answer» Correct Answer - D 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.