Paragraph: f( x )= array lr 3 x^2+12 x-1 & -1 x 2 37-x, & 2 x 3 array . Question: Which of the following statements are correct? 1. f(x) is continuous at x=2 . 2. f(x) attains greatest value at x=2 . 3. f ( x ) is differentiable at x =2 . Select the correct answer using the code given below:

Step-by-step solution

For continuity at x=2 . RHL _ x 2⁺ f(x)= _ x 2⁺ 37-x=37-2=3 LHL aligned & _ x 2⁻ f(x)= _ x 2⁻ 3 x^2+12 x-1 & =3(2)^2+12 2-1=12+24-1=35 & f(2)=3 4+12 2-1=12+24-1=35 aligned So , RHL = LHL f(x) is continuous at x=2 . For differentiability at x=2 . aligned & LHD = _ x 2⁻ f(x)-f(2) x-2 = _ h 0 f(2-h)-f(2) 2-h-2 & = _ h 0 3(2-h)^2+12(2-h)-(12+24-1) -h & = _ h 0 3 h^2-24 h -h = _ h 0 24-3 h=24 & RHD = _ x 2⁺ f(x)-f(2) x-2 = _ h 0 f(2+h)-f(2) 2+h-2 & = _ h 0 37-2-h-35 h =-1 aligned LHD _ RHD f(x) is not differentiable at