Question

Find k value if f(x)=(x-k)^4 is minimum when x=3

Answer 1

the answer is in the above picture
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Answer 2

The value of k = 3.

Given:

• Function f(x) = (x - k)⁴
• Function is minimum at x = 3

To find: Value of k.

Step by step explanation:

→ Basic Identities used

• (a - b)² = a² + b² - 2ab
• (a - b) (a - b) = (a - b)²
• (a - b)⁴ = (a - b)² * (a - b)²
  

We know that the function is minimum at x = 3.
So, substituting the value of x = 3 in the function and solving it.

⇒ f(x) = (x - k)⁴
⇒ f(3) = (3 - k)⁴
⇒ 0 = (3 - k)² * (3 - k)²
⇒ [(3)² + (k)² - 2*3*k] * [(3)² + (k)² - 2*3*k] = 0
⇒ (9 + k² - 6k) * (9 + k² - 6k) = 0
⇒ (9 + k² - 6k) =         0            
                            (9 + k² - 6k)
⇒ (9 + k² - 6k) = 0
⇒ k² - 6k + 9 = 0
⇒ k² - 3k - 3k + 9 = 0
⇒ k (k - 3) - 3 (k - 3) = 0
⇒ (k - 3) * (k - 3) = 0
⇒  k - 3 =         0            
                     ( k - 3 )
⇒ k - 3 = 0
⇒ k = 3

∴ The value of k = 3.

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