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In the mean value theorem, $f(b) - f(a) = (b - a)f'(c) $ if $a = 4$, $b = 9$ and $f(x) = \sqrt x $ then the value of $c$ is
$8$
$5.25$
$4$
$6.25$
Solution
(d) $f(x) = \sqrt x $
$\therefore \,\,f(a) = \sqrt 4 = 2,$ $f(b) = \sqrt 9 = 3$ ; $f'(x) = \frac{1}{{2\sqrt x }}$
Also, $f'(c) = \frac{{f(b) – f(a)}}{{b – a}} = \frac{{3 – 2}}{{9 – 4}} = \frac{1}{5}$
$\therefore $ $\frac{1}{{2\sqrt c }} = \frac{1}{5}$ ==> $c = \frac{{25}}{4} = 6.25$.
Similar Questions
Let $f, g:[-1,2] \rightarrow R$ be continuous functions which are twice differentiable on the interval $(-1,2)$. Let the values of $f$ and $g$ at the points $-1.0$ and $2$ be as given in the following table:
$x=-1$ | $x=0$ | $x=2$ | |
$f(x)$ | $3$ | $6$ | $0$ |
$g(x)$ | $0$ | $1$ | $-1$ |
In each of the intervals $(-1,0)$ and $(0,2)$ the function $(f-3 g)^{\prime \prime}$ never vanishes. Then the correct statement(s) is(are)
$(A)$ $f^{\prime}(x)-3 g^{\prime}(x)=0$ has exactly three solutions in $(-1,0) \cup(0,2)$
$(B)$ $f^{\prime}(x)-3 g^{\prime}(x)=0$ has exactly one solution in $(-1,0)$
$(C)$ $f^{\prime}(x)-3 g^{\prime}(x)=0$ has exactly one solution in $(0,2)$
$(D)$ $f^{\prime}(x)-3 g^{\prime}(x)=0$ has exactly two solutions in $(-1,0)$ and exactly two solutions in $(0,2)$