Let \( f(x)=|3-x|+(3+x) \) be a function, then which of the following statements are TRUE?
(A) \( f(x) \) is continuous at \( x=3 \)
(B) \( f(x) \) is continuous at \( x=-3 \)
(C) \( f(x) \) is differentiable at \( x=3 \)
(D) \( f(x) \) is differentiable at \( x=-3 \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A), (B) and (D) only
Since |3 − x| = 3 − x for x ≤ 3 and x − 3 for x > 3, we can remove the modulus:
\( \displaystyle f(x)=\begin{cases}6, & x\le 3\\2x, & x\gt 3\end{cases} \)
At x = 3:
\( \displaystyle \lim_{x\to 3^-}f(x)=6 \)
\( \displaystyle \lim_{x\to 3^+}f(x)=2(3)=6 \)
\( f(3)=6 \)
\( Lf'(3)=0 \) (slope of 6)
\( Rf'(3)=2 \) (slope of 2x)
Near x = −3, f(x) = 6, a constant.
So f is continuous at x = 3 but its left and right derivatives there are 0 and 2, and near x = −3 it is the constant 6 (continuous, with derivative 0).
Checking the statements
(A) \( f(x) \) is continuous at \( x=3 \) — Correct. Both one-sided limits and f(3) equal 6.
(B) \( f(x) \) is continuous at \( x=-3 \) — Correct. f(x) = 6 near x = −3.
(C) \( f(x) \) is differentiable at \( x=3 \) — Incorrect. The left derivative is 0 and the right derivative is 2.
(D) \( f(x) \) is differentiable at \( x=-3 \) — Correct. f(x) = 6 near x = −3, so f′(−3) = 0.