Let $$f(x)$$ satisfy all the conditions of Lagrange's mean value theorem in [$$0$$,$$2$$] . If $$f(0)=0$$ and f′(x)≤$$12$$ for all x in [$$0$$,$$2$$], then

Question

Let $$f(x)$$ satisfy all the conditions of Lagrange's mean value theorem in [$$0$$,$$2$$] . If $$f(0)=0$$ and f′(x)≤$$12$$ for all x in [$$0$$,$$2$$], then

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By using lagrange's mean value theorem ,$$f(b)$$−$$f(a)b$$−$$a=f$$′(x)⇒$$f(2)$$−$$f(0)2$$−$$0=f$$′(x)⇒$$f(2)$$−$$02=f$$′(x)integrating$$ on both sides⇒$$f(x)=f(2)2x+c$$ $$($$∵$$f(0)=0$$⇒$$c=0)∴$$f(x)=f(2)2x$$ ……(1) Also, f′(x)≤$$12$$⇒$$f(2)2$$≤$$12From$$  Eq. (1), |$$f(x)$$|$$=f(2)2x=f(2)2$$|x|≤$$12$$|x| In interval [$$0$$,$$2$$], for maximum value of x take $$x=2$$|$$f(2)$$|≤$$12$$⋅$$2$$⇒|$$f(2)$$|≤$$1hence$$ fx≤$$1$$

$Let f (x) satisfy all the conditions of Lagrange's mean value theorem in [0, 2] . If$ $f (0) = 0 and |f^{'} (x)| \leq \frac{1}{2} for all x in [0, 2], then$

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Let f(x) satisfy all the conditions of Lagrange's mean value theorem in [0,2] . If f(0)=0 and f′(x)≤12 for all x in [0,2], then

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$Let f (x) satisfy all the conditions of Lagrange's mean value theorem in [0, 2] . If$ $f (0) = 0 and |f^{'} (x)| \leq \frac{1}{2} for all x in [0, 2], then$