Consider the function, f(x) = | x – 2 | + | x – 5 |, x ∈ R. Statement 1 : f '(3) = 0 Statement 2 : f is continuous in [2, 5], differentiable in (2, 5) and f(2) = f(5).

Engineering

Mathematics

Important Points on Derivability

Question

Consider the function, f(x) = | x – 2 | + | x – 5 |, x ∈ R.

Statement 1 : f '(3) = 0

Statement 2 : f is continuous in [2, 5], differentiable in (2, 5) and f(2) = f(5).

Statement 1 is true, Statement 2 is false.

Statement 1 is false, Statement 2 is true.

Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.

Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 2.

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f(x) = | x – 2 | + | x – 5 | , x ∈ R

$f (x) = {\begin{cases} - 2 x + 7, x < 2 \\ 3, 2 \leq x \leq 5 \\ 2 x - 7, x > 5 \end{cases}$

It is clear that f(x) is continuous in R and

$(- \infty, 2) \cup (2, 5) \cup (5, \infty)$

$∴$ Statement 2 is correct.

Statement 1 is also correct but Statement 2 is not the correct explanation of Statement 1.