Divide 10 into two parts such that the sum of | vedclass

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(A) Let the two parts be $x$ and $(10 - x)$.Define the function $f(x) = 2x + (10 - x)^2$.To find the minimum,differentiate with respect to $x$:$f'(x)

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$9, 1$

Vedclass · Answer

(A) Let the two parts be $x$ and $(10 - x)$.Define the function $f(x) = 2x + (10 - x)^2$.To find the minimum,differentiate with respect to $x$:$f'(x) = 2 + 2(10 - x)(-1) = 2 - 20 + 2x = 2x - 18$.Set $f'(x) = 0$ to find the critical point:$2x - 18 = 0 \implies x = 9$.Find the second derivative to check for minima:$f''(x) = 2$.Since $f''(9) = 2 > 0$,the function has a local minimum at $x = 9$.The two parts are $x = 9$ and $10 - x = 1$.

Divide $10$ into two parts such that the sum of twice the first part and the square of the second part is minimum. The two parts are:

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