# Gradients, tangent planes and extrema

> The gradient collects first partial derivatives and determines directional derivatives through a dot product.

Course: [University calculus and multivariable methods](https://studybrix.com/en/courses/university-calculus/index.md)
Level: university
Prerequisites: Partial derivatives and vector dot products.

## Notes

### Gradient

The gradient collects first partial derivatives and determines directional derivatives through a dot product.

### Tangent plane

Linearization at a differentiable point uses f(a,b)+fₓ(a,b)(x−a)+fᵧ(a,b)(y−b).

### Stationary points

A zero gradient is necessary for an interior differentiable extremum, not sufficient.

### Hessian test

In two variables, positive determinant and positive fₓₓ identify a strict local minimum; negative determinant identifies a saddle.

## Practice questions

### 1. For f(x,y)=x²+3y², find fₓ at (2,1).

Answer: 4

Explanation: Differentiate with respect to x to obtain 2x, then substitute x=2.

### 2. Write the tangent plane to z=x²+y² at (1,2,5).

Sample answer / marking guidance: z=5+2(x−1)+4(y−2), equivalently z=2x+4y−5.

Explanation: Credit the correct base height and both gradient coefficients.

### 3. For f(x,y)=x²+3y², find the directional derivative at (1,1) in direction (3/5,4/5).

Answer: 6

Explanation: The gradient is (2,6). Its dot product with (3/5,4/5) is 6/5+24/5=6.

### 4. What kind of stationary point is the origin for f=x²−y²?

A. A strict local maximum
B. A point with no partial derivatives
C. A saddle point
D. A strict local minimum

Answer: A saddle point

Explanation: The function increases along the x-axis and decreases along the y-axis.

### 5. Find the determinant of the Hessian of f=2x²+3xy+5y².

Answer: 31

Explanation: The Hessian is [[4,3],[3,10]]; its determinant is 40−9=31.

### 6. Why should the direction vector be normalized before calculating a directional derivative?

Sample answer / marking guidance: A directional derivative is conventionally the rate per unit distance in that direction. Without normalization, the dot product is scaled by the vector length.

Explanation: Credit the unit-distance interpretation and the scaling effect.

### 7. If a stationary point has Hessian determinant zero, what does the ordinary second-derivative test conclude?

A. It must be a minimum
B. It must be a maximum
C. It must be a saddle
D. It is inconclusive

Answer: It is inconclusive

Explanation: Higher-order terms or another argument may be needed.

### 8. Give a differentiable function whose gradient is zero at the origin but that has no local extremum there.

Sample answer / marking guidance: For example f(x,y)=x²−y² has gradient zero at the origin but takes both positive and negative values arbitrarily close to it.

Explanation: Credit a valid example and evidence that neither minimum nor maximum holds.

## Further reading

- https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/
- https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/

Original AI-assisted educational draft; not independently teacher-reviewed. References support further study, not copied exercises, affiliation or curriculum accreditation.