Definite Integration Ques 40
- If $\lim _{t \rightarrow a} \frac{\int _a^{t} f(x) d x-\frac{(t-a)}{2}{f(t)+f(a)}}{(t-a)^{3}}=0$,
the degree of polynomial function $f(x)$ at most is
0
1
3
2
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Solution:
- Given, $\lim _{t \rightarrow a} \frac{\int _a^{t} f(x) d x-\frac{(t-a)}{2}{f(t)+f(a)}}{(t-a)^{3}}=0$
Using L’Hospital’s rule, put $t-a=h$
$$ \begin{array}{ll} \Rightarrow \quad \lim _{h \rightarrow 0} \frac{\int _a^{a+h} f(x) d x-\frac{h}{2}{f(a+h)+f(a)}}{h^{3}} & =0 \\ \Rightarrow \quad \lim _{h \rightarrow 0} \frac{f(a+h)-\frac{1}{2}{f(a+h)+f(a)}-\frac{h}{2}{f^{\prime}(a+h) }}{3 h^{2}} & =0 \end{array} $$
Again, using L’ Hospital’s rule,
$\lim _{h \rightarrow 0} \frac{f(a+h)-\frac{1}{2} f(a+h)-\frac{1}{2} f(a+h)-\frac{h}{2} f^{\prime \prime}(a+h)}{6 h}=0$
$$ \begin{aligned} \Rightarrow & & \lim _{h \rightarrow 0} \frac{-\frac{h}{2} f^{\prime \prime}(a+h)}{6 h} & =0 \\ \Rightarrow & & f^{\prime \prime}(a) & =0, \forall a \in \mathbb{R} \end{aligned} $$
$\Rightarrow f(x)$ must have maximum degree 1 .
BETA
