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Mathematics
10.10.2021 12:47
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Find the smallest possible value of a quadratic equation

Find the smallest possible value of a quadratic equation
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skiddymark3ox93za
skiddymark3ox93za
4,7(37 marks)

I'm assuming that you mean to find the minimum of a parabola, i.e. the minimum of a function defined as

f(x) = ax^2+bx+c,\quad a,b,c \in\mathbb{R},\quad a\neq 0

To find the minimum of a function, we have to find a point x_0 such that

f'(x_0) = 0,\quad f''(x_0) 0

The first derivative is

f'(x) = 2ax+b \implies f'(x)=0 \iff x = \dfrac{-b}{2a}

The second derivative is

f''(x) = 2a

So, a parabola has a minimum only if a0 (otherwise, the parabola is concave down and it has no lower bound). In that case, the minimum has coordinates

x = \dfrac{-b}{2a},\quad y = f\left(\dfrac{-b}{2a}\right) = a\left(\dfrac{-b}{2a}\right)^2 + b\left(\dfrac{-b}{2a}\right) + c = \dfrac{4ac-b^2}{4a}

karellopez96
karellopez96
4,8(45 marks)

D. (4,2)

Step-by-step explanation:

Find the point that is in the dark region

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