​ If f(x) = ∫(5+|1-t|)dt, x ∈ [0,x], x>2, 5x+1, x≤2, then ​(1) f(x) is not continuous at x = 2 (2) f(x) is everywhere differentiable

Question

If f(x) = \(\begin{cases} \int\limits^x_0(5+|1-t|)dt, & x>2 \\ 5x+1, & x\leq2 \end{cases}\), then

{ ∫(5+|1-t|)dt,x ∈ [0,x] x > 2, 5x+1, x ≤ 2

(1) f(x) is not continuous at x = 2

(2) f(x) is everywhere differentiable

(3) f(x) is continuous but not differentiable at x = 2

(4) f(x) is not differentiable at x = 1

Answer 1

Answer is: (3) f(x) is continuous but not differentiable at x = 2

ƒ(2) = ƒ(2^–) = 5 × 2 + 1 = 11

⇒ continuous at x = 2

Clearly differentiable at x = 1

Lf’(2) = 5

Rf’(2) = 6

⇒ not differentiable at x = 2