First Principles AI

Accedmychevron_right11thchevron_rightmathchevron_rightTheory Of Quadratic Functionschevron_rightExercise 3.1

(i) $x^2+6x+13$

\boxed{\min f(x)=4\text{ at }x=-3}

Step 1: Coefficient of ( x^2 ) is positive (1), so the function has a minimum value.

Step 2: Complete the square:

f(x) = x^2 + 6x + 13

f(x) = (x^2 + 6x + 9) + 13 - 9

f(x) = (x + 3)^2 + 4

Step 3: Since ( (x + 3)^2 \geq 0 ), minimum occurs when:

x + 3 = 0 \Rightarrow x = -3

Step 4: Minimum value:

f(-3) = 0 + 4 = 4

\boxed{\text{Minimum value} = 4 \text{ at } x = -3}