Answer:
the function is continuous but not differentiable in x=3
Step-by-step explanation:
for f(x) defined as
x²+5 for x≤3
3*x+5 for x>3
then for f to be continuous then
when x→a , lim f(x) = f(a)
thus
from the left
when x→3⁻ , lim f(x) = lim x²+5 = 3²+5 = 14
from the right
when x→3⁺ , lim f(x) = lim 3*x+5 = 3*3+5 = 14
and f(3) = 3²+5 = 14
since both limits converge and are equal to f(3), then the function is continuous in 3
Nevertheless, the function is not differentiable in 3 since the curve is not smooth in f(3) ( it has an sharp change in f(3) ) , then the function is not differentiable in x=3
to prove it , when x=3 and
when h→0⁻ , f'(x=3)= lim [f(x+h)-f(x)] /h = lim [(x+h)²+5 - (x² + 5)]/h = lim [2*x*h+h²] /h = lim [2*x+h] = 2*3+0 = 6
when h→0⁺ , f'(x=3)= lim [f(x+h)-f(x)] /h = lim [3*(x+h)+5 - (3*x + 5)]/h = lim [3*h] /h = lim 3 = 3
since the limits do not converge →the limit does not exist → the derivative does not exist → the function is not differentiable in x=3
