f(x)=⎧⎪
⎪
⎪
⎪
⎪
⎪
⎪⎨⎪
⎪
⎪
⎪
⎪
⎪
⎪⎩(1+|sinx|)a|sinx| ,−π6<x<0b ,x=0etan2xtan3x ,0<x<π6
Since, f(x) is continuous at x=0
⇒limh→0f(0−h)=f(0)=limh→0f(0+h)
Now,
limh→0f(0−h)=limh→0(1+|sin(−h)|)a|sin(−h)|
=limh→0(1+sinh)asinh
=e(1+sinh−1)asinh=ea
So,
ea=b
Also, limh→0f(0+h)=limh→0etan2htan3h
=limh→0etan2h2h×3htan3h×23
=e2/3
⇒ea=b=e2/3
⇒a=23
∴3(a+logb)=3(23+23)=4