8.05e-g Stationary points and tangent planes Quiz: OCR Further Maths, Unit 5

The 20 questions

1. What first-derivative conditions must hold at a stationary point of $z = f(x,y)$?

   • $f_{xx} = 0$ and $f_{yy} = 0$
   • $f_x + f_y = 0$
   • $f_x = 0$ and $f_y = 0$
   • $f_x = f_y \neq 0$

2. How is the discriminant $D$ defined for classifying stationary points of $z = f(x,y)$?

   • $f_{xx} + f_{yy} - f_{xy}$
   • $f_{xx} f_{yy} - (f_{xy})^2$
   • $f_{xx} f_{yy} + (f_{xy})^2$
   • $(f_{xy})^2 - f_{xx} f_{yy}$

3. If $D > 0$ and $f_{xx} > 0$ at a stationary point, what type of point is it?

   • Saddle point
   • Local minimum
   • Local maximum
   • Inconclusive point

4. If $D > 0$ and $f_{xx} < 0$ at a stationary point, what type of point is it?

   • Local maximum
   • Local minimum
   • Inconclusive point
   • Saddle point

5. What type of stationary point occurs when the discriminant $D$ is strictly negative?

   • Local minimum
   • Point of inflection
   • Local maximum
   • Saddle point

6. What does a discriminant value of zero indicate when classifying a stationary point?

   • Saddle point
   • Local maximum
   • Test is inconclusive
   • Local minimum

7. Which vector is normal to the surface $z = f(x,y)$ at the point $(x_0, y_0, z_0)$?

   • $(f_x, f_y, -1)$
   • $(1, 1, f_z)$
   • $(f_x, f_y, 1)$
   • $(f_x, f_y, 0)$

8. What is the general equation of the tangent plane to $z = f(x,y)$ at $(x_0,y_0,z_0)$?

   • $z = f_x x + f_y y$
   • $z - z_0 = f_x(x-x_0) f_y(y-y_0)$
   • $z - z_0 = f_{xx}(x-x_0) + f_{yy}(y-y_0)$
   • $z - z_0 = f_x(x-x_0) + f_y(y-y_0)$

9. What is the partial derivative $f_x$ for the function $f(x,y) = x^3 y^2 + 4x$?

   • $3x^2 y^2$
   • $3x^2 + 4y$
   • $3x^2 y^2 + 4$
   • $2x^3 y + 4$

10. Where is the stationary point located for the surface $z = x^2 + y^2 - 4x + 6y$?

    • $(4, -6)$
    • $(-2, 3)$
    • $(2, 3)$
    • $(2, -3)$

11. What type of stationary point is at the origin for the surface $z = x^2 + y^2$?

    • Local maximum
    • Saddle point
    • Inconclusive point
    • Local minimum

12. What type of stationary point is at the origin for the surface $z = x^2 - y^2$?

    • Saddle point
    • Local minimum
    • Local maximum
    • Inconclusive point

13. For $z = x^2 y$, what are the values of $(f_x, f_y)$ at the point $(2, 3)$?

    • $(12, 12)$
    • $(4, 12)$
    • $(6, 4)$
    • $(12, 4)$

14. What is the equation of the tangent plane to $z = x^2 + y^2$ at $(1,1,2)$?

    • $2x + 2y - z = 2$
    • $x + 2y - z = 1$
    • $2x + 2y + z = 6$
    • $x + y - z = 0$

15. Find a normal vector to $z = x^2 + y^2$ at the point $(1, 2, 5)$.

    • $(1, 2, -1)$
    • $(1, 4, -1)$
    • $(2, 4, 1)$
    • $(2, 4, -1)$

16. Under what condition on smooth surfaces does Clairaut's theorem state that $f_{xy} = f_{yx}$?

    • Surface is linear
    • Derivatives are continuous
    • Derivatives are zero
    • Discriminant is positive

17. How many stationary points exist on the surface $z = x^3 - 3x + y^2$?

    • 1
    • 4
    • 2
    • 3

18. What type of stationary point is located at $(1, 0, -2)$ for $z = x^3 - 3x + y^2$?

    • Inconclusive point
    • Local minimum
    • Local maximum
    • Saddle point

19. What vector gives the direction of the normal line to $z = xy$ at $(2,3,6)$?

    • $(3, 2, 1)$
    • $(3, 2, -1)$
    • $(2, 3, -1)$
    • $(6, 6, -1)$

20. What geometric condition makes the tangent plane to $z = f(x,y)$ horizontal at a point?

    • Boundary point
    • Asymptotic point
    • Inflection point
    • Stationary point

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