for what value of c will the funciton x^6+9x^4+26x^2 +C have real roots?

Question

I have the answer of -2, I just don't know how they got it.

Answer 1

This is a continuous function. Plugging in 1 we get 41+C. Plugging in -1 we get the same, 41+C. Plugging in 0 we get C.

 

Choose C < 0 and C > -41. Then by the intermediate value theorem of Calculus, as the function goes from 41+C at x=-1 to C at 0 it crosses the x-axis, and it crosses it again as it goes from 0 to 41+C at x=1. That gives (at least) two real roots. 

 

If you mean plural "values", i.e. you want to characterize all of the values of C that work, then certainly C > 0 doesn't work because all the other summands are perfect squares, hence are non-negative, so adding them gives something >= 0, and then adding C will take us above 0. So C <= 0 is an upper bound. We show that any value of C <  0 works. Let C be < 0. Then choose x to be the sixth (positive) root of -C. For example, if C = -729 then take x to be 3. We conclude

 

x⁶+9x⁴+26x²+C >  x6 + C >= 0,  and likewise taking x to be the negative square root of -C, so the intermediate value theorem argument works again, giving at least two roots.

 

The function has one real root if C = 0.

 

Therefore 

 

the function will have one or more real roots if and only if C <= 0

the function will have exactly one real root if and only if C = 0

 

It is also easy to show that the function has zero, one, or exactly two real roots by considering the first derivative, which is positive to the right of 0 and negative to the left of 0, hence there are no local extrema other than the minimum at 0.

Answer 2

Note that x^6 + 9x^4 + 26x^2 has all positive coefficients and x only has even powers.  Thus, this is an even polynomial whose values range from 0 to infinity and is monotone increasing on (0, infinity) and monotone decreasing on (-infinity, 0).  Thus, if C is greater than 0, the function will be positive everywhere, and if C < 0, the function will be negative at 0 and near 0 and go to positive infinity as x goes to +- infinity, and have 2 roots, and if C = 0, there is a root at 0.

 

There are roots if C <= 0.