5 (1) Let / C R be a non-degenerate open bounded interval, let c e / and let f; - R be a function. Suppose that f is differentiable at c. (1) Prove...

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4.5.5 (1)Let / C R be a non-degenerate open bounded interval, let c e / and let f; - R be afunction. Suppose that f is differentiable at c.(1) Prove that if f'(c) > 0, then there is some $ > 0 such that (c - 6, c + 8) $ 1,that x E.(c - 6, c] implies f (x) < f(c), and that x E [c, c + 6) implies f(x) > f(c).f'(c) = limx-+af(x)-f(c)x-CIt is given f'(c) > 0 so limx-,af(x)-f(c)X-C30VE, 35 > 0 wheref(x)-f(QX-G- f'(c) < = when lx - cl < 8This implies f'(c) - E <f(x)-f(c)X-C<f'(c) +EF(x)-f (c)>f'(c) - E>0It follows thatf(x)-f(c)>0X-Cf(x) - f(c) >0wherec <x<c+6 - f(x) >f(c)f(x) - f(c) < 0wherec- 6 <x<c- f(x) <f(c)