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  • Let f be a differentiable function on the open interval (a, b). Which of the following statements must be true ? 1. f is continuous on the closed interval [a, b]. II. f is boun

Let f be a differentiable function on the open interval (a, b).
Which of the following statements must be true ?
1. f is continuous on the closed interval [a, b].
II. f is bounded on the open interval (a, b).
1II. If \mathrm{ a<a_1<b_1<b and f\left(a_1\right)<0<f\left(b_1\right)}, then there is a number ' c ' such that \mathrm{ a_1<c<b_1 and f(c)=0}

Option: 1

\text { I and II only. }


Option: 2

\text { I and III only. }


Option: 3

\text { II and III only. }


Option: 4

\text { Only III. }


Answers (1)

best_answer

\mathrm{ I \,\, and\, \, II are\, \, false, The function, f(x)=\frac{1}{x}, 0<x<1, .}
is a counter example. Statement III is true. Apply the intermediate value theorem to f on the closed interval \mathrm{ \left[a_1, b_1\right] }

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SANGALDEEP SINGH

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