Show that `f(x)={12 x-13 ,\ \ \ if\ xlt=3 2x^2+5,\ \ \ \ \ if\ x >3` is differentiable at `x=3` . Also, find `f^(prime)(3)` .

Explanation

Given function,
`f(x)={{:(12x-13,if xle3),(2x^2+5,if x>3):}`
We know that function is differentiable at x, if LHD at `x =` RHD at `x` .

Now,
LHD of `f(x)` at `x=3`,
`lim_(h->0)(12(3-h)-13-(12(3)-13))/((3-h)-3)=12`
RHD of `f(x)` at `x=3`,
`lim_(h->0)(2(3+h)^2+5-(2(3)^2+5))/((3+h)-3)=12`
Since LHD = RHD, `f(x)` is differentiable at `x=3 `.
`f'(3) = 12`