Question

Let f(x)={x^2-16x,x<9}{12√x,x>=9}if f continuous at x = 9 ? Determine whether f is  differentiable at x = 9 . If so, find the value of the derivative there.

Answer 1

The plot of this function is

lower part of the plot corresponds to parabola, upper part - to the square root. This function is not continuous at $x=9$ since

$lim_{x \to 9-0} = 81- 16 \cdot 9 = -63$

$lim_{x \to 9+0} = 12 \cdot 3 = 36$

(this conclusion can be drawn from the plot as well).

The function is not continuous at $x=9$ , therefore it is not differentiable at this point.