Question

The minimum value of x²-2x+5 is

Answer 1

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Here is your answer...✌

We have ,

f(x) = x² - 2x + 5

At the point of maximum or minimum

f'(x) = 0

⇒ 2x - 2 = 0

⇒ x = 2/2

⇒ x = 1

Now,

f''(x) = 2 , which is positive

Therefore we get a minimum value of f(x) at x = 1

f(1) = (1)² - 2(1) + 5

= 1 - 2 + 5

= -1 + 5

= 4

Hence,

The minimum value of x² - 2x + 5 is 4

Answer 2

Final answer : 4
Another Method :

Let f(x) = x^2-2x+5

$= {x}^{2} - 2x + 1 + 4 \\ = {(x - 1)}^{2} + 4$
Since, (x-1)^2 >=0 for all x belongs to R.
At x = 1 , minimum value of
f(x) exists.
Therefore,
min f(x) = 0 + 4
= 4