Convex and concave curves and points of inflection Quiz

1. On which sections is a curve convex?

• A. Where $\frac{dy}{dx}>0$
• B. Where $\frac{d^2y}{dx^2}>0$
• C. Where $\frac{d^2y}{dx^2}<0$
• D. Where $y>0$

2. $y=x^4$ has $\frac{d^2y}{dx^2}=12x^2$, which is zero at $x=0$. Which statement is correct?

• A. The origin is a point of inflection because the second derivative is zero
• B. The curve is concave for $x<0$
• C. The curve is convex only for $x>0$
• D. The origin is not a point of inflection because the second derivative does not change sign

3. Find the $x$-coordinate of the point of inflection of $y=x^3-3x^2$.

• A. $x=0$
• B. $x=2$
• C. $x=1$
• D. $x=-1$

4. $\frac{d^2y}{dx^2}=2-x$. For which values of $x$ is the curve concave?

• A. $x<2$
• B. $x>2$
• C. $x>-2$
• D. $x<-2$

5. At which value of $x$ is the gradient of $y=x^3-3x^2$ least?

• A. $x=1$
• B. $x=0$
• C. $x=2$
• D. $x=3$