Can you sketch the following curves?

1. A continuous curve with exactly one point with zero gradient and exactly two zeros.

2. A continuous curve with exactly two points with zero gradient and exactly two zeros.

3. A continuous curve with exactly one point with zero gradient and exactly one zero.

4. A continuous curves with exactly two points with zero gradient and exactly one zero.

Can you give examples of algebraic equations that satisfy each of the conditions above?

Extension:

Prove that in a much larger catalogue you could construct examples of continuous curves with exactly $N$ points of zero gradient and exactly $M$ zeros for any non-negative whole numbers $N$ and $M$.

1. A continuous curve with exactly one point with zero gradient and exactly two zeros.

2. A continuous curve with exactly two points with zero gradient and exactly two zeros.

3. A continuous curve with exactly one point with zero gradient and exactly one zero.

4. A continuous curves with exactly two points with zero gradient and exactly one zero.

Can you give examples of algebraic equations that satisfy each of the conditions above?

Extension:

Prove that in a much larger catalogue you could construct examples of continuous curves with exactly $N$ points of zero gradient and exactly $M$ zeros for any non-negative whole numbers $N$ and $M$.