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5. Continuity and Differentiation
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Let $f (x)$ and $g (x)$ are two function which are defined and differentiable for all $x \ge x_0$. If $f (x_0) = g (x_0)$ and $f ' (x) > g ' (x)$ for all $x > x_0$ then
A
$f (x) < g (x)$ for some $x > x_0$
B
$f (x) = g (x)$ for some $x > x_0$
C
$f (x) > g (x)$ only for some $x > x_0$
D
$f (x) > g (x)$ for all $x > x_0$
Solution
Consider$\phi (x) = f (x) – g (x)$==>$\phi '(x) = f ' (x) – g ' (x)$
$\phi (x)$ is also continuous and derivable in $[x_0, x]$
using $LMVT$ for $Q(x)$ in $[x_0, x]$
$\phi '(x) = \frac{{\varphi (x) – \varphi ({x_0})}}{{x – {x_0}}}$.
since $\phi ' (x) = f ' (x) – g ' (x)$ are $f ' (x) – g ' (x) > 0$
$\phi ' (x) > 0$
hence$\phi (x) – \phi (x_0) > 0$
$\phi (x) > \phi (x_0)$
$f (x) – g (x) > 0$
are $f (x_0) = f (x_0) – f (x_0) = 0$ ==>$(D)$
Standard 12
Mathematics
