# find the range of $x$ on which $f$ is decreasing, where $f(x)=int_0^{x^2-x}e^{t^2-1}dt$

Mathematics Asked by Steven Lu on December 4, 2020

I want to find the range of $$x$$ on which $$f$$ is decreasing, where
$$f(x)=int_0^{x^2-x}e^{t^2-1}dt$$

Let $$u=x^2-x$$, then $$frac{du}{dx}=2x-1$$, then $$f'(x)=frac{d}{dx}int_0^{x^2-x}e^{t^2-1}dt=frac{du}{dx}frac{d}{du}int_0^{x^2-x}e^{t^2-1}dt=(2x-1)e^{x^4-2x^3+x^2-1}$$

Since $$e^{x^4-2x^3+x^2-1}>0$$ for all $$xin Bbb R$$ and $$2x-1<0iff x. $$f$$ is decreasing on $$(-infty,frac{1}{2})$$.

Furthermore, $$f$$ is increasing on $$(frac{1}{2},infty)$$, $$f$$ is differentiable at $$x=frac{1}{2}$$, and $$f'(frac{1}{2})=0$$, $$f$$ attains its minimum value at $$x=frac{1}{2}$$.

Am I right?

$$f(x)=int_{0}^{x^2-x} e^{t^2-1} dt implies f'(x)= (2x-1) e^{(x^2-x)^2-1} >0 ~if~ x>1/2$$. Hence $$f(x)$$ in increasing for $$x>1/2$$ and decreasing for $$x<1/2$$. Yes you are right. there is a min at $$x=1/2$$. This one point does not matter, you may also say that $$f(x)$$ is increasing in $$[1/2,infty)]$$ and decreasing on 4(-infty, 1/2]$. Note: whether a function increasing or decreasing is decided by two points (not one). For instance, $$x_1>x_2 leftrightarrows f(x_1) > f(x_2).$$ If $$f(x)$$ is decreasing. Answered by Z Ahmed on December 4, 2020 Everything is fine ! A little bit more can be said: $$f$$ is strictly decreasing on $$(-infty,frac{1}{2}]$$ and $$f$$ is strictly increasing on $$[frac{1}{2}, infty).$$ Answered by Fred on December 4, 2020 ## Add your own answers! ## Related Questions ### Does the Riemannian distance function obey the Leibniz rule? 1 Asked on November 24, 2021 by asaf-shachar ### Non Case Analysis Proof of Triangle Inequality for Hamming Distance 1 Asked on November 24, 2021 by henry-powell ### A non-circular argument that uses the Maclaurin series expansions of$sin x$and$cos x$to show that$frac{d}{dx}sin x = cos x$1 Asked on November 24, 2021 ###$|int_a^b f(x)dx| leq |int_a^b |f(x)| dx| $? 1 Asked on November 24, 2021 ### Number of permutations of the letters$a, b, c, d$such that$b$does not follow$a$, and$c$does not follow$b$, and$d$does not follow$c$2 Asked on November 24, 2021 ### Diagonalizability of an operator 1 Asked on November 24, 2021 by gba ### What is the probability for$x$to be positive only? 2 Asked on November 24, 2021 by amateurashish ### A simple proof of Sylow theorem for abelian groups 1 Asked on November 24, 2021 ### Question about the proof of the index theorem appearing in Milnor’s Morse Theory 1 Asked on November 24, 2021 by doggerel ### Why can 2 uncorrelated random variables be dependent? 3 Asked on November 24, 2021 ### How to solve$intfrac{1}{sqrt {2x} – sqrt {x+4}} , mathrm{dx} $? 3 Asked on November 24, 2021 by flinn-bella ### Integral:$int dfrac{dx}{(x^2-4x+13)^2}$? 9 Asked on November 24, 2021 by user809843 ### What is the intersection of inductive definable subsets of a real closed field? 2 Asked on November 24, 2021 ### How can I prove that the following are happening:$lnBig(1+frac{1}{x}Big)=frac{1}{x}+oBig(frac{1}{x}Big)$? 2 Asked on November 24, 2021 by andvld ### Having (x,y,t) determine if two persons were near for more than 5 minutes 1 Asked on November 24, 2021 ### Understanding notation and meaning of uniform convergence of a power series 1 Asked on November 24, 2021 ### Proving L’Hospital’s rule 1 Asked on November 24, 2021 by blackthunder ### Is it possible to calculate$x$and$y\$?

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