If the sum of two numbers is 3,find the maxim | vedclass

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(A) Let the first number be $(3 - x)$ and the second number be $x$.Define the function $f(x) = (3 - x)x^2 = 3x^2 - x^3$.To find the critical points,ca

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$4$

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(A) Let the first number be $(3 - x)$ and the second number be $x$.Define the function $f(x) = (3 - x)x^2 = 3x^2 - x^3$.To find the critical points,calculate the first derivative:$f'(x) = 6x - 3x^2$.Set $f'(x) = 0$:$3x(2 - x) = 0 \Rightarrow x = 0$ or $x = 2$.Find the second derivative to check for maxima:$f''(x) = 6 - 6x$.Evaluate at $x = 2$:$f''(2) = 6 - 6(2) = -6 Since the second derivative is negative,$x = 2$ gives the maximum value.Maximum value = $(3 - 2) 	imes 2^2 = 1 	imes 4 = 4$.

If the sum of two numbers is $3$,find the maximum value of the product of the first number and the square of the second number.

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