If f(x)= array cc (1+| x|)^ a | x| , & - / 6 < x < 0 b & x=0 2 x e^ 3 x , & 0 < x < / 6 array . is continuous at x=0 , find the values of a and b .

Step-by-step solution

We have, _ x 0⁻ f(x) aligned =& _ x 0 1+| x| ^ a | x| =& _ x 0 | x| a | x| =e^ a and _ x 0⁺ f(x)= _ x 0 e^ 2 x 3 x =& _ x 0 2 x 2 x 3 x 3 x 2 3 =& e^ 2 / 3 aligned For f(x) to be continuous at x=0 , we must have aligned & _ x 0⁻ f(x) &= _ x 0⁺ f(x) & &=f(0) & e^ a &=e^ 2 / 3 =b & a &=2 / 3 and & b &=e^ 2 / 3 aligned