Let f : I R → I R be defined as f ( x ) = ∣ x ∣ + x 2 − 1 . The total number of points at which f attains either a local maximum or a local minimum is

Question

Let f : I R → I R be defined as f ( x ) = ∣ x ∣ + ​ x 2 − 1 ​ . The total number of points at which f attains either a local maximum or a local minimum is

TestHub · Accepted Answer

f ( x ) = ∣ x ∣ + ​ x 2 − 1 ​ = ⎩ ⎨ ⎧ ​ − x + x 2 − 1 , x < − 1 − x − x 2 + 1 , − 1 ≤ x ≤ 0 x − x 2 + 1 , 0 < x < 1 x 2 + x − 1 , x ≥ 1 ​ ∴ f ′ ( x ) = ​ 2 x − 1 − 2 x − 1 − 2 x + 1 2 x + 1 ​ , , , , ​ x < − 1 − 1 ≤ x ≤ 0 0 < x < 1 x > 1 ​ Critical points are 2 1 ​ , 2 − 1 ​ , − 1 , 0 and 1 . We observe at five points f ′ ( x ) changes its sign ∴ There are 5 points at which either local maximum or local minimum.

Mathematics - Application of Derivative Question with Solution | TestHub

Doubts & Discussion