Please answer completely thank you. a) What are critical numbers of the function? b) What test do you use to find the critical numbers? c) What are points of inflection? d) What test do you use to find the points of inflection? 2. Find the critical numbers of the function. a)f(x) =x^4 + x^3 + x^2 + 1 Solution a) Critical numbers are the values of x for which f\'(x) =0 or f\'(x) does not exist b) to find critical points we check for f\'(x) = 0 or for the points where f(x) does not exist that is , we find values for which f(x) is not continuous c) points of inflection are the points at which the sign of curvature changes d) we check for points where f\'\'(x) =0 although this is a necessary conditions , we check lowest higher order derivative , for change in sign of curvature 2) f\'(x) = d/dx ( x^4 + x^3 + x^2 + 1) = 4x^3 + 3x^2 + 2x f\'(x) =0 => 4x^3 + 3x^2 + 2x =0 => x (4x^2+3x+2) =0 since 4x^2 + 3x + 2 =0 does not have real roots it has only one critical point x=0.

1. Please answer completely thank you. a) What are critical numbers of the function? b) What test
do you use to find the critical numbers? c) What are points of inflection? d) What test do you use
to find the points of inflection? 2. Find the critical numbers of the function. a)f(x) =x^4 + x^3 +
x^2 + 1
Solution
a)
Critical numbers are the values of x for which f'(x) =0 or f'(x) does not exist
b)
to find critical points we check for f'(x) = 0 or for the points where f(x) does not exist that is , we
find values for which f(x) is not continuous
c)
points of inflection are the points at which the sign of curvature changes
d)
we check for points where f''(x) =0 although this is a necessary conditions , we check lowest
higher order derivative , for change in sign of curvature
2)
f'(x) = d/dx ( x^4 + x^3 + x^2 + 1)
= 4x^3 + 3x^2 + 2x
f'(x) =0
=>
4x^3 + 3x^2 + 2x =0
=>
x (4x^2+3x+2) =0
since 4x^2 + 3x + 2 =0 does not have real roots
it has only one critical point x=0