Write a function with the given properties

Question

1) Write a function with a critical point at -1 and f(-1)=2 and f''(-1)=4.

2) Write a function with a saddle point at -1 and f(-1)=2 and f''(-1)=4.

3) Write a function with a local min at -1 and f''(-1)=0.

Answer 1

1) A quadratic can solve this part:

f(x)=Ax²+ Bx + C

f'(x)=2Ax + B

f"(x)=2A

therefore 2A = 4 => A=2

4x+B=0 when x = -1 => B=4

and 2x² + 4x + C = 2 when x=-1 => C=4

2) A saddle point usually requires a function of two variables...so I am not sure how to answer this part

3) Usually when the 2nd derivative is zero at a critical point there is a horizontal tangent...so f(x)=(x+1)³ has f'(-1)=0 and f"(-1)=0...but x=-1 is NOT a local minimum.....but f(x)=(x+1)⁴ should serve your purpose