Question

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

Statement I f'(4) = 0

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

(a) Statement I is false, Statement II is true

(b) Statement II is true, Statement II is true; Statement II is a correct explanation of Statement I

(c) Statement I is true, Statement II is true; Statement II is not a correct explanation of Statement I

(d) Statement I is true, Statement II is false

Answer 1

Correct Option (c) Statement I is true, Statement II is true; Statement II is not a correct explanation of Statement I

Explanation:

Given A function f such that

f(x) = |x - 2| + |x - 5|

To discuss Continuity and differentiability interval (2, 5)of f

in 

Now, we can draw the graph of f very easily

From the above graph we can analyse all the required things.

Statement I f'(4) = 0

it is obviously clear that f is constant around x = 4.

hence f'(4) = 0.

Hence, Statement I is correct.

Statement ll

It can be clearly seen that

(i) f is continuous, ∀x ∈ [2, 5]

(ii) f is differential, ∀x ∈ (2, 5)

(iii) f'(2) = f(5) = 3

Hence, Statement ll is also correct but obviously not a correct explanation of Statement I.