# 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^3
diff(f(x),x)

# Part (b): Find where the function is increasing and decreasing:

solve((4*x^3 + 12*x^2)>0,x)

solve((4*x^3 + 12*x^2)<0,x)

# Part (c): Find the local max and local min if any:
f(-3)

# Part (d): Take the second derivative to find where the function is concave up and concave down:  
diff(f(x),x,2)

solve ((12*x^2 + 24*x)>0,x)

solve ((12*x^2 + 24*x)<0,x)

# Part (e): Find the infection points to show where the function changes concavity:
f(-2)

f(0)

# Part (f): Calculate the limits of the function as x goes to +/- infinity:

limit(f,x=+infinity)

limit(f,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)