Is #f(x)=(1-e^x)/x^2# increasing or decreasing at #x=2#?
2 Answers
The function
Explanation:
Calculate the first derivative and look at the sign
We need
Here,
Therefore,
When
As
graph{(y-(1-e^x)/(x^2))(y-1000(x-2))=0 [-4.93, 12.85, -4.124, 4.765]}
Explanation:
#"to determine if a function f(x) is increasing/decreasing at"#
#"x = a evaluate "f'(a)#
#• " if "f'(a)>0" then f(x) is increasing at x = a"#
#• " if "f'(a)<0" then f(x) is decreasing at x = a"#
#"differentiate "f(x)" using the "color(blue)"quotient rule"#
#"given "f(x)=(g(x))/(h(x))" then"#
#f'(x)=(h(x)g'(x)-g(x)h'(x))/(h(x))^2larr" quotient rule"#
#g(x)=1-e^xrArrg'(x)=-e^x#
#"h(x)=x^2rArrh'(x)=2x#
#rArrf'(x)=(-x^2e^x-2x(1-e^x))/(4x^2)#
#rArrf'(2)=(-4e^2-4+4e^2)/16=-1/4#
#"since "f'(2)<0" then f(x) is decreasing at x = 2"#
graph{(1-e^x)/(x^2) [-10, 10, -5, 5]}