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Math Help - Intermediate Value Theorem

  1. #1
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    Intermediate Value Theorem

    Hi all,

    How can I use the Intermediate Value Theorem to show that the equation
    x^4+x^3+x^2+x+\frac{1}{4}=0
    has at least two real solutions.. Thanks.
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  2. #2
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    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.
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  3. #3
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    Hey,
    I do not understand what you mean by "Prove it and use the fact that the polynomial becomes very big and positive when x grows in absolute value".. can you elaborate a bit on this ? thanks.
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  4. #4
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    Let the polynomial be denoted by f(x). Prove that f(-0.5) < 0 by computing f(-0.5). Further, f(x)\to+\infty when 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 x_1 < -0.5 and an x_2 > 0 such that f(x_1) > 0 and f(x_2) > 0. Thus, f(x) changes sign from x_1 to -0.5 and from -0.5 to x_2.
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