What is the minimum?

Algebra Level 2

Find the minimum value of the expression below for real x x .

x 2 − 12 x + 40 x^2-12x+40


The answer is 4.

Anuj Shikarkhane by Anuj Shikarkhane , 19, India

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2 solutions

Chew-Seong Cheong
Jun 15, 2018

f ( x ) = x 2 − 12 x + 40 = x 2 − 12 x + 36 + 4 = ( x − 6 ) 2 + 4 Note that ( x − 6 ) 2 ≥ 0 ≥ 4 \begin{aligned} f(x) & = x^2 - 12x + 40 \\ & = x^2 - 12x + 36 + 4 \\ & = {\color{#3D99F6}(x-6)^2} + 4 & \small \color{#3D99F6} \text{Note that }(x-6)^2 \ge 0 \\ & \ge \boxed{4} \end{aligned}

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Hana Wehbi
Jun 15, 2018

To find min or max, we take the first derivative and set equal to zero as follows:

f ’ ( x ) = 2 x − 12 = 0 ⟹ x = 6 ⟹ y = 36 − 72 + 40 = 4 ⟹ the minimum occurs at y = 4 f’(x)=2x-12=0\implies x=6\implies y=36-72+40=4\implies \text { the minimum occurs at } y=4

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