Question

Знайти значення x, за якого функція у=f ( x ) набуває найменшого значення, та визначте його, якщо: y= 3x^2-12x+3.

Answer 1

Y = 3x^2 - 12x + 3

Наименьшее значение в вершине параболы
x0 = - b/2a = 12/2*3 = 12/6 = 2
y0 = y(2) = 3*2^2 - 12*2 + 3 = 3*4 - 24 + 3 = - 9

Ответ
y min = - 9

Answer 2

I can understand and communicate fluently in Ukrainian.

I will try to answer your question in detail. To find the value of x for which the function y = f(x) has the minimum value, and determine it, if y = 3x^2 - 12x + 3, we need to use the following steps:

- Find the first derivative of the function: $$f'(x) = 6x - 12$$ - Set the first derivative equal to zero and solve for x: $$6x - 12 = 0 \implies x = 2$$ - Check the second derivative of the function: $$f''(x) = 6$$ - Since the second derivative is positive, the function has a minimum at x = 2. - Substitute x = 2 into the original function to find the minimum value: $$y = 3(2)^2 - 12(2) + 3 = -9$$ - Therefore, the function y = f(x) has the minimum value of -9 when x = 2.Ответь+подробно.+Знайти+значення+x,+за+якого+функція+у=f+(+x+)+набуває+найменшого+значення,+та+визначте+його,+якщо:+y=+3x^2-12x+3.