Question

At x=3, the function given by f(x)={x²,,​x<3 and 6x−9, x≥3​ is:

(A) undefined.
(B) continuous but not differentiable.
(C) differentiable but not continuous.
(D) neither continuous nor đifferentiable.
(E) both continuous and differentiable.

Answer 1

The function is both continuous and differentiable at x=3.

Step-by-step explanation:

The function given by f(x) is defined differently for x<3 and x≥3. Let's evaluate the function at x=3. For x<3, f(x)=x², so f(3)=3²=9. For x≥3, f(x)=6x-9, so f(3)=6(3)-9=9. Since the function is defined and equal to 9 at x=3 for both cases, the function is continuous at x=3. It is also differentiable at x=3 because both parts of the function have derivatives that exist at x=3. Therefore, the function is both continuous and differentiable at x=3, so the correct answer is (E) both continuous and differentiable.