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Math Help - Prove that f is strictly increasing

  1. #1
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    Prove that f is strictly increasing

    f:[0,∞) → ℝ, f(x) = e^(x^2)

    In order to prove that f is strictly increasing, do I have to show that f'(x) is always positive?

    I think f'(x)=[(x^2)e^(x^2)]/e + 2xe^(x^2)lne
    I could have gotten that horribly wrong, so correct me if I have.

    Then how to prove that this is strictly positive?

    Thanks.
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  2. #2
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    Quote Originally Posted by feyomi View Post
    f:[0,∞) → ℝ, f(x) = e^(x^2)

    In order to prove that f is strictly increasing, do I have to show that f'(x) is always positive?

    I think f'(x)=[(x^2)e^(x^2)]/e + 2xe^(x^2)lne
    I could have gotten that horribly wrong, so correct me if I have.

    Then how to prove that this is strictly positive?

    Thanks.

    Sorry just to add to that. Is it enough to say that each term in f'(x) is positive since xϵ[0,∞) and since e>0 and so f'(x)>0, thus f is strictly increasing?
    Could you also please just check my answer for f'(x)?

    Thanks
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  3. #3
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    Quote Originally Posted by feyomi View Post
    f:[0,∞) → ℝ, f(x) = e^(x^2)

    In order to prove that f is strictly increasing, do I have to show that f'(x) is always positive?

    I think f'(x)=[(x^2)e^(x^2)]/e + 2xe^(x^2)lne
    I could have gotten that horribly wrong, so correct me if I have.

    Then how to prove that this is strictly positive?

    Thanks.
    f(x) = e^{x^2}

    f'(x) = 2x e^{x^2}

    for x > 0 , f'(x) > 0
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  4. #4
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    Hahaha of course. Silly me.
    Thank you.
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