First derivative test

Question

f(x)=x³+x⁴ first derivative test 

Answer 1

f(x) = x³+x⁴

 

f'(x) = 3x²+4x³ = x²(3+4x) 

 

                        = 0 when x = 0 or -3/4  ← critical points

 

When x < -3/4, f'(x) < 0  So, f is decreasing.

 

When -3/4 < x < 0, f'(x) > 0   So, f is increasing

 

When x > 0, f'(x) > 0,  So, f is increasing

 

Conclusion:  Since f'(x) changes sign from negative to positive when x = -3/4, there is a relative min when x = -3/4

 

                   Since f'(x) doesn't change sign when x = 0, there is no relative extremum when x = 0.