Trying to find the right answer to this..

Let f (x) = – (x – 2) 3 + 4 Now ( x – 2) 2 ≥ 0 ∀ x ∊ R ⇒ – (x - 2) 2 ≥ 0 ∀ x ∊ R ⇒ – (x – 2) 2 + 4 ≤ 4 ∀ x ∊ R ∴ maximum value of f (x) is 4. It has no minimum value.

Application of Derivatives

Question

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Find the maximum and minimum values, if any, of the following functions without using the derivatives:
– (x – 2 )² + 4

Answer

Let f (x) = – (x – 2)³ + 4
Now ( x – 2)² ≥ 0 ∀ x ∊ R
⇒ – (x - 2)² ≥ 0 ∀ x ∊ R ⇒ – (x – 2)² + 4 ≤ 4 ∀ x ∊ R
∴ maximum value of f (x) is 4. It has no minimum value.