Can you solve this without Calculus?

Let's use the A.M-G.M. inequality

We know that $A.M \geq G.M$ if each of the individual terms are positive .

$\dfrac{x + \frac{25}{x}}{2} \geq \sqrt{ x\cdot \frac{25}{x}} \\ \Rightarrow x + \frac{25}{x} \geq 10$

Q.E.D

$x+ \frac{25}{x} = \frac{1}{x}(x-5)^2+10 \geq 10$

It should be stated 'minimum positive value', but anyways.

$25$ has the factors $1,5,25$

which when taken as $x$ give $26, 10, 26$

Hence, $10$

The expression is the sum of the straight line y+x and the hyperbola y = 25/x. We note that the smallest value for x is when the "curves" intersect; i.e. x= 5, Then the sum is 5 + 25/5 = 10. Ed Gray

i use the derivative formula, is it the calculus stuff that i shouldn't use?

f(x) = x + 25/x

f'(x) = (x^2 - 25)/x^2

f'(x) = 0

(x^2 - 25)/x^2 = 0

x^2 - 25 = 0

x^2 = 25

x = 5 since x > 0

put it in the function so

5 + 25/5 = 10