Hi all,
How can I use the Intermediate Value Theorem to show that the equation
$\displaystyle x^4+x^3+x^2+x+\frac{1}{4}=0$
has at least two real solutions.. Thanks.
Looking at the plot in WolframAlpha, it looks like the value at -0.5 is negative. Prove it and use the fact that the polynomial becomes very big and positive when x grows in absolute value. You can easily find values of x for which the polynomial is positive.
Let the polynomial be denoted by f(x). Prove that f(-0.5) < 0 by computing f(-0.5). Further, $\displaystyle f(x)\to+\infty$ when $\displaystyle x\to\pm\infty$. (Since you are studying the Intermediate Value Theorem, which uses continuous functions, you are supposed to know what a limit is.) In fact, it is not necessary to invoke limits; it is easy to find an $\displaystyle x_1 < -0.5$ and an $\displaystyle x_2 > 0$ such that $\displaystyle f(x_1) > 0$ and $\displaystyle f(x_2) > 0$. Thus, f(x) changes sign from $\displaystyle x_1$ to -0.5 and from -0.5 to $\displaystyle x_2$.