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Math Help - Differentiabl Prob.

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    Differentiabl Prob.

    True or False: If f and g are differentiable on [a,b] and |f'(x)| \le 1 \le |g'(x)| \forall x \in (a,b), then |f(x) - f(a)| \le |g(x) - g(a)| \forall x \in [a,b]

    Since f,g are differentiable on [a,b] they are also continuous on [a,b]....All i can derive from this problem so far. Help a brother continue. Thank you.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Endowed View Post
    True or False: If f and g are differentiable on [a,b] and |f'(x)| \le 1 \le |g'(x)| \forall x \in (a,b), then |f(x) - f(a)| \le |g(x) - g(a)| \forall x \in [a,b]

    Since f,g are differentiable on [a,b] they are also continuous on [a,b]....All i can derive from this problem so far. Help a brother continue. Thank you.
    Suppose that |f(x)-f(a)|>|g(x)-g(a)| for some x\in(a,b) then, clearly \left|\frac{f(x)-f(a)}{x-a}\right|>\left|\frac{g(x)-g(a)}{x-a}\right| and since by assumption f,g are differentiable on [a,x] we know that the mean value theorem guarantees us that |f'(c)|=\left|\frac{f(x)-f(a)}{x-a}\right|>\left|\frac{g(x)-g(a)}{x-a}\right|=|g'(e)| for some c,e\in(a,x)\subseteq (a,b)...so..
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