How can I find the intervals $a<x<a+1$,where a is zero or an integer which contain the real roots of the equation $x^4-12x+5=0$

Question

How can I find the intervals $a<x<a+1$,where a is zero or an integer which contain the real roots of the equation $x^4-12x+5=0$

did I need to solve the given equation.then it would be a very lengthy process.is there any short process?can anyone help me?thanks.

Answer 1

Let $f(x) = x^4 -12x + 5$. $$f(\text{negative number}) > 0$$ $f(0) = 5$, $f(1) = -6$, $f(2) = -3$, $f(3) = 50$.

Hence, there are roots between $[0,1]$ and $[2,3]$.

Further $f'(x) = 4x^3 - 12$. Hence, the function is increasing for $x>\sqrt[3]{3}$ and decreasing for $x< \sqrt[3]{3}$.

Hence, there are only two real roots; one in the interval $[0,1]$ and the other in the interval $[2,3]$.