# Kristen Shine# Final Examf(x)=3*x^3-5*x^2-x+8fprime(x)=diff(f(x),x)fprime(x)
9*x^2 - 10*x - 1
plot(f(x),-10,10,ymin=-10,ymax=10)
# To find critical points, we set the derivative equal to 0 and solve.solve(fprime(x)==0,x)x1=-1/9*sqrt(34)+5/9x2=1/9*sqrt(34)+5/9n(x1)n(x2)y1=f(x1)y2=f(x2)
# To find point of inflection, find second derivative of f, set it equal to 0, and solve.fdoublep(x)=diff(fprime(x))solve(fdoublep==0,x)x3=(5/9)y3=n(f(x3))# (x3,y3) is the functions inflection point.
[x == (5/9)]
„#Where is the function f(x) increasing and decreasing?# Interval - (-infinity,x1),(x1,x2),(x2,infinity)# Plug numbers into derivative function from each intervalfprime(-3)fprime(1/3)fprime(3)# From -infinity to x1, the function f(x) is increasing.# From x1 to x2, the function f(x) is decreasing.# From x2 to infinity, the function f(x) is increasing.# When the value of fprime(x) is positive, this means the f(x) function is increasing. Same with negative and decreasing.
File:Docstring:
110
-10/3
50
# To find the y coordinates, take f of x1 and x2.y1=f(x1)n(y1)y2=f(x2)n(y2)# (x1,y1) is the functions local max# (x2,y2) is the functions local min
8.04734456316659
4.78393115699802
# plotting function with points -plot(f(x),-5,5,ymin=-10,ymax=10)+point((x1,y1),size=30)+point((x2,y2),size=30)+point((x3,y3),size=30)