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Math Help - Continuity - continuous functions

  1. #1
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    Continuity - continuous functions

    Dont quite understand what my lecturer is wanting me to do with this question.. Don't know if i am trying to/thinking its more complicated then it actually is.

    <br />
F(x) = x^3 + 2x+cos^2x<br />

    (a) Explain why there is a number x such that f(x) = 0
    (b) Explain why there is only one value of x such that f(x) = 0

    Thanks,
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  2. #2
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    Quote Originally Posted by Robb View Post
    Dont quite understand what my lecturer is wanting me to do with this question.. Don't know if i am trying to/thinking its more complicated then it actually is.

    <br />
F(x) = x^3 + 2x+cos^2x<br />

    (a) Explain why there is a number x such that f(x) = 0
    (b) Explain why there is only one value of x such that f(x) = 0

    Thanks,
    F(x) = x^3 + 2x + \cos^2{x} is continuous.

    F(-1) < 0 and F(0) > 0 ... by the Intermediate Value Theorem, there must exist an x = c , -1 < c < 0, such that F(c) = 0.

    F'(x) = 3x^2 + 2 - 2\cos{x}\sin{x} = 3x^2 + 2 - \sin(2x) > 0 for all x ... F(x) is always increasing and therefore, can only have a single zero.
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  3. #3
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    Quote Originally Posted by Robb View Post
    <br />
F(x) = x^3 + 2x+cos^2x
    (a) Explain why there is a number x such that f(x) = 0
    (b) Explain why there is only one value of x such that f(x) = 0
    Can you see that F(-1)<0\;\&\;F(0)>0?
    Use the IVT to conclude that there is a root between -1 & 0.

    Is it true that F'(x)>0? What does that tell us?
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