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APPLICATION OF DERIVATIVES — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAPPLICATION OF DERIVATIVES3 Marks
Question
Let f be a function defined on [a, b] such that f′(x) > 0, for all $\text{x}\in(\text{a},\text{b}).$ Then prove that f is an increasing function on (a, b).
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Answer

Since, f'(x) > 0 on (a, b)
Then, f is a differentiating function (a,b)
Also, every differentiating function is continuous,
Therefore, f is continuous on [a, b]
Let x1, x2 $\in(\text{a},\text{b})$ and x2 > x1 then by LMV
theorem, there exists c $\in(\text{a},\text{b})$ s.t.
$\text{f}'(\text{c})=\frac{\text{f(x}_2-\text{f(x}_1))}{\text{x}_2-\text{x}_1}$
$\Rightarrow\ \text{f(x}_2)-\text{f(x}_1)=\text{(x}_2-\text{x}_1)\text{f}'\text{(c)}$
$\Rightarrow\ \text{f(x}_2)-\text{f(x}_1)>0\text{ as x}_2>\text{x}_1\text{and f}'\text{(x)}>0$
$\Rightarrow\ \text{f(x}_2)>\text{f(x}_2)$
$\therefore\ \text{x}_1<\text{x}_2\ \Rightarrow\ \text{f(x}_1<\text{f(x}_2)$
Thwrefore, f is an increasing function.

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