




Plot the function:
Step 1
No singularity
Step 2
Calculation of f ' and f"
f(x) = x^{3} 3x^{2}+1
f '(x) = 3x^{2}  6x
f"(x) = 6x  6
Step 3
f '(x) = 3x^{2}  6x
= 3x(x2)
f '(x) = 0 for x = 0 or x = 2 (critical points)
f(0) = 1 and f(2) = 3 (critical values)
f '(x) > 0 for x < 0 and x > 2
f is increasing for x < 0 and x > 2
f '(x) < 0 for 0 < x < 2
f is decreasing for 0 < x < 2
Step 4
f"(x) = 6x  6
f"(x) = 0 for x = 1 (inflection point)
f"(x) < 0 for x < 1
f is concave down for x < 1
f"(x) > 0 for x > 1
f is concave up for x > 1