Question

if the differentiable function f(x) on (a,b) is constant then
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Answer 1

Answer:

Theb solution for differential equation is $\int_{a}^{b} f^{2}(x)-f^{2}(a) d x$

Step-by-step explanation:

A monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order.

Let $\mathrm{f}^{-1}(\mathrm{x})=\mathrm{t} \Rightarrow \mathrm{x}=\mathrm{f}(\mathrm{t})$

$2 \int_{f(a)}^{f(b)} x\left(b-f^{-1}(x)\right) d x=2 \int_{a}^{b}(b-t) f(t) f^{\prime}(t) d t$

$\begin{aligned}&=\left[(b-t)\left(f(t)^{2}\right)^{2}\right]_{a}^{b}+\int_{a}^{b}(f(t))^{2} d t \\&=-(b-a)(f(a))^{2}+\int_{a}^{b}(f(t))^{2} d t \\&=-\int_{a}^{b}(f(a))^{2} d t+\int_{a}^{b}(f(t))^{2} d t\end{aligned}$

$=\int_{a}^{b}\left(f^{2}(t)-\left(f^{2}(a)\right)\right) d t$

Now replace the value of $t \rightarrow X$

then $\int_{a}^{b} f^{2}(x)-f^{2}(a) d x$

your question is incomplete but most probably your full question was

"If $\mathrm{f}(\mathrm{x})$ is a monotonic and differentiable function on ${a,b}$ then $2 \int_{\mathrm{f}(\mathrm{a})}^{\mathrm{f}(\mathrm{b})} \mathrm{x}\left(\mathrm{b}-\mathrm{f}^{-1}(\mathrm{x})\right) \mathrm{dx}$is equal to?"