# Curve Sketching: use f(x)=x^4+4*x^3 to answer the following questions :# Part (a): Find the derivative of the f(x):f(x)=x^4+4*x^3diff(f(x),x)
4*x^3 + 12*x^2
# Part (b): Find where the function is increasing and decreasing:solve((4*x^3+12*x^2)>0,x)
[[x > -3, x < 0], [x > 0]]
solve((4*x^3+12*x^2)<0,x)
[[x < -3]]
# Part (c): Find the local max and local min if any:f(-3)
-27
# Part (d): Take the second derivative to find where the function is concave up and concave down: diff(f(x),x,2)
12*x^2 + 24*x
solve((12*x^2+24*x)>0,x)
[[x < -2], [x > 0]]
solve((12*x^2+24*x)<0,x)
[[x > -2, x < 0]]
# Part (e): Find the infection points to show where the function changes concavity:f(-2)
-16
f(0)
0
# Part (f): Calculate the limits of the function as x goes to +/- infinity:limit(f,x=+infinity)
x |--> +Infinity
limit(f,x=-infinity)
x |--> +Infinity
# Part (g): Graph the functions: plot(x^4+4*x^3,-5,5,ymin=-30,ymax=40)+plot(4*x^3+12*x^2,-5,5,color='red',ymin=-30,ymax=40)+plot(12*x^2+24*x,-5,5,color='green',ymin=-30,ymax=40)