Invisible Function

Algebra Level 4

f ( x 2 + x + 1 ) = 2 − 6 x − 7 x 2 − 2 x 3 − x 4 f(x^2 + x+1) = 2-6x-7x^2-2x^3-x^4

Let f f be the polynomial function, satisfying the constraint above. What is the maximum value of f f ?


The answer is 11.

Worranat Pakornrat by Worranat Pakornrat , 36, Thailand

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1 solution

f ( x 2 + x + 1 ) = 2 − 6 x − 7 x 2 − 2 x 3 − x 4 Given = 2 − ( x 4 + 2 x 3 + 3 x 2 + 2 x + 1 + 4 x 2 + 4 x + 4 ) + 5 = 7 − 4 ( x 2 + x + 1 ) − ( x 2 + x + 1 ) 2 Replace x 2 + x + 1 with x ⟹ f ( x ) = 7 − 4 x − x 2 \begin{aligned} f(x^2+x+1) & = 2-6x-7x^2-2x^3-x^4 & \small \color{#3D99F6} \text{Given} \\ & = 2 - (x^4+2x^3+3x^2+2x+1 + 4x^2+4x+4) + 5 \\ & = 7- 4(x^2+x+1) - (x^2+x+1)^2 & \small \color{#3D99F6} \text{Replace }x^2+x+1 \text{ with }x \\ \implies f(x) & = 7-4x-x^2 \end{aligned}

To find the maximum of f ( x ) f(x) , f ′ ( x ) = − 4 − 2 x f'(x) = -4-2x . Therefore, f ′ ( x ) = 0 f'(x) = 0 , when x = − 2 x=-2 . We note that f ′ ′ ( x ) = − 2 < 0 f''(x) = -2 < 0 . This implies that max ⁡ ( f ( x ) ) = f ( − 2 ) = 7 − 4 ( − 2 ) − ( − 2 ) 2 = 11 \max (f(x)) = f(-2) = 7-4(-2) - (-2)^2 = \boxed{11} .

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