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Math Help - Discrete Math. Urgent help please.

  1. #1
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    Discrete Math. Urgent help please.

    Give a proof of the following implication: Let x be an arbritrary real number.

    If 2x^2 + 2/x^2 = pi, then x^pi + x^(log10) = e^pi*pi^e
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  2. #2
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    Hello Juancd08
    Quote Originally Posted by Juancd08 View Post
    Give a proof of the following implication: Let x be an arbritrary real number.

    If 2x^2 + 2/x^2 = pi, then x^pi + x^(log10) = e^pi*pi^e
    I think this is all to do with a false statement implying a statement which may be true or false. See the third and fourth lines of the Truth Table: Discrete mathematics/Logic/Page 2 - Wikibooks, collection of open-content textbooks.

    y=2x^2+\frac{2}{x^2}

    \Rightarrow \frac{dy}{dx}=4x-\frac{4}{x^3}=0 when x = \pm 1, \pm i

    \frac{d^2y}{dx^2}= 4+\frac{12}{x^4}>0 when x = \pm 1

    So if x is real, the minimum value of 2x^2+\frac{2}{x^2} is 4, and therefore 2x^2+\frac{2}{x^2} = \pi has no real solutions.

    So if p(x) is the propositional function: 2x^2+\frac{2}{x^2}=\pi and q(x) is the propositional function x^{\pi} + x^{\log 10} = e^{\pi}\cdot\pi^e:

    \forall x \in \mathbb{R},\, p \Rightarrow q, since p is always false

    i.e. for all real values of x, 2x^2+\frac{2}{x^2} = \pi \Rightarrow x^{\pi} + x^{\log 10} = e^{\pi}\cdot\pi^e .

    Grandad
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