Find the minimum

Algebra Level 3

f ( x ) = x 4 + 4 x 3 + 6 x 2 + 4 x + 8 \LARGE f(x)=x^4+4x^3+6x^2+4x+8

What's the minimum value of f ( x ) f(x) ?


The answer is 7.

Md Mehedi Hasan by Md Mehedi Hasan , 22, Bangladesh

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2 solutions

Anirudh Sreekumar
Nov 25, 2017

f ( x ) = x 4 + 4 x 3 + 6 x 2 + 4 x + 8 = ( x + 1 ) 4 + 7 Since , ( x + 1 ) 4 ≥ 0 ⟹ f ( x ) ≥ 7 Thus minimum value of f ( x ) = 7 at x = − 1 \begin{aligned} f(x)&=x^4+4x^3+6x^2+4x+8\\ &=(x+1)^4+7\\ &\text{Since ,}\\ &(x+1)^4\geq0\\ \implies &f(x)\geq7\\ &\text{Thus minimum value of } f(x)=7 \text{ at } x=-1\end{aligned}

3 Helpful 0 Interesting 0 Brilliant 0 Confused
Syed Hamza Khalid
Nov 25, 2017

I basically used trial and error. I started out with 1, 2 and then I tried 0.Finally I tried out negative numbers and when I reached -1. I noticed that it was the smallest value since after substituting it. I get 7.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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