# Data Science for Engineers | Week 5

**Session: JAN-APR 2024**

**Course Name: Data Science for Engineers**

**Course Link: Click Here**

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**These are NPTEL Data Science for Engineers Assignment 5 Answers**

**Q1. Which of the following statements is/are not TRUE with respect to the multi variate optimization?I – The gradient of a function at a point is parallel to the contoursII – Gradient points in the direction of greatest increase of the functionIII – Negative gradients points in the direction of the greatest decrease of the functionIV – Hessian is a non-symmetric matrix**

I

II and III

I and IV

III and IV

**Answer: I and IV**

**Q2. The solution to an unconstrained optimization problem is always the same as the solution to the constrained one.**

True

False

**Answer: False**

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**These are NPTEL Data Science for Engineers Assignment 5 Answers**

**Q3. Gradient-based algorithm methods compute**

only step length at each iteration

both direction and step length at each iteration

only direction at each iteration

none of the above

**Answer: both direction and step length at each iteration**

**Q4. For an unconstrained multivariate optimization given f(x¯), the necessary second order condition for x¯∗ to be the minimizer of f(x¯) is**

∇2f(x¯∗) must be negative definite.

∇2f(x¯∗) must be positive definite.

∇f(x¯∗)=0

”(x∗)>0

**Answer: ∇2f(x¯∗) must be positive definite.**

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**These are NPTEL Data Science for Engineers Assignment 5 Answers**

Consider an optimization problem minx1,x2∈Rf(x1,x2)=x21+4×22−2×1+8×2.

**Q5. Which among the following is the stationary point for f(x1,x2)?**

(0, 0)

(1, −1)

(−1, −1)

(−1, 1)

**Answer: (1, −1)**

**Q6. Find the eighen values corresponding to Hessian matrix of f.**

1, −1

1, 1

2, 8

0, 2

**Answer: 2, 8**

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**These are NPTEL Data Science for Engineers Assignment 5 Answers**

**Q7. Find the minimum value of f.**

0

−5

−1

1

**Answer: −5**

**Q8. Now, in order to find the minimum value of f subject to the constraint x1+2×2=7,what should be the first order condition for x∗¯ to be a minimizer of f(x1,x2)?**

a) 2x∗1+2=λ

−8x∗2−8=2λ

x∗1+2x∗2=7

b) −2x∗1+2=λ

−8x∗2−8=2λ

x∗1+2x∗2=7

c) 2x∗1−2=−λ

8x∗2+8=−2λ

x∗1+2x∗2=7

d) −2x∗1+2=−λ

−8x∗2−8=−2λ

x∗1+2x∗2=7

**Answer: b) −2x∗1+2=λ −8x∗2−8=2λ x∗1+2x∗2=7**

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**These are NPTEL Data Science for Engineers Assignment 5 Answers**

**Q9. What is the minimum value of f(x1,x2) subject to the above mentioned constrained?**

−5

−1

27

0

**Answer: 27**

**Q10. Find the maximum value of f(x,y)=49−x2−y2 subject to the constraints x+3y=10.**

49

46

59

39

**Answer: 39**

**Q11. Consider an optimization problem minx1,x2x2−xy+y2 subject to the constraints2x+y≤1x+2y≥2x≥−1Find the lagrangian function for the above optimization problem.**

L(x,y,μ1,μ2,μ3)=x2−xy+y2+μ1(2x+y−1)+μ2(2−x−2y)+μ3(−x−1)

L(x,y,μ1,μ2,μ3)=x2−xy+y2+μ1(2x+y−1)+μ2(x+2y−2)+μ3(−x−1)

L(x,y,μ1,μ2,μ3)=x2−xy+y2+μ1(2x+y−1)+μ2(x+2y−2)+μ3(x+1)

L(x,y,μ1,μ2,μ3)=x2−xy+y2+μ1(1−2x−y)+μ2(2−x−2y)+μ3(−x−1)

**Answer: L(x,y,μ1,μ2,μ3)=x2−xy+y2+μ1(2x+y−1)+μ2(2−x−2y)+μ3(−x−1) **

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**These are NPTEL Data Science for Engineers Assignment 5 Answers**

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**Session: JULY-DEC 2023**

**Course Name: Data Science for Engineers**

**Course Link: Click Here**

**These are NPTEL Data Science for Engineers Assignment 5 Answers**

**Q1. The values of μ1,μ2 and μ3 while evaluating the Karush-Kuhn-Tucker (KKT) condition with all the constraints being inactive are**

μ1=μ2=μ3=1

μ1=μ2=μ3=0

μ1=μ3=0,μ2=1

μ1=μ2=0,μ3=1

**Answer: μ1=μ2=μ3=0**

**Q2. Gradient based algorithm methods compute**

only step length at each iteration

both direction and step length at each iteration

only direction at each iteration

none of the above

**Answer: both direction and step length at each iteration**

**These are NPTEL Data Science for Engineers Assignment 4 Answers**

**Q3. The point on the plane x+y−2z=6 that is closest to the origin is**

(0,0,0)

(1,1,1)

(−1,1,2)

(1,1,−2)

**Answer: (1,1,−2)**

**Q4. Find the maximum value of f(x,y)=49−x2−y2 subject to the constraints x+3y=10.**

49

46

59

39

**Answer: 39**

**These are NPTEL Data Science for Engineers Assignment 4 Answers**

**Q5. The minimum value of f(x,y)=x2+4y2−2x+8y subject to the constraint x+2y=7 occurs at the below point:**

(5,5)

(−5,5)

(1,5)

(5,1)

**Answer: (5,1)**

**Q6. Which of the following statements is/are NOT TRUE with respect to the multi variate optimization?I – The gradient of a function at a point is parallel to the contoursII – Gradient points in the direction of greatest increase of the functionIII – Negative gradients points in the direction of the greatest decrease of the functionIV – Hessian is a non-symmetric matrix**

I

II and III

I and IV

III and IV

**Answer: I and IV**

**These are NPTEL Data Science for Engineers Assignment 4 Answers**

**Q7. The solution to an unconstrained optimization problem is always the same as the solution to the constrained one.**

True

False

**Answer: False**

**Q8. A manufacturer incurs a monthly fixed cost of $7350 and a variable cost,C(m)=0.001m3−2m2+324m dollars. The revenue generated by selling these units is, R(m)=−6m2+1065m. How many units produced every month (m) will generate maximum profit?**

(m)=46

(m)=90

(m)=231

(m)=125

**Answer: (m)=90**

**These are NPTEL Data Science for Engineers Assignment 4 Answers**

**Q9. Consider an optimization problem minx1,x2 x2−xy+y2 subject to the constraints 2x+y≤1 x+2y≥2 x≥−1Find the lagrangian function for the above optimization problem.**

L(x,y,μ1,μ2,μ3) = x2 − xy + y2 + μ1(2x + y − 1) + μ2(2 − x − 2y) + μ3( −x − 1)

L(x,y,μ1,μ2,μ3) = x2 −xy + y2 + μ1(2x + y − 1) + μ2(x + 2y − 2)) + μ3( −x − 1)

L(x,y,μ1,μ2,μ3) = x2 − xy + y2 + μ1(2x + y − 1) + μ2(x + 2y − 2)) + μ3(x + 1)

L(x,y,μ1,μ2,μ3) = x2 − xy + y2 + μ1(1 − 2x − y) + μ2(2 − x − 2y) + μ3( − x − 1)

**Answer: L(x,y,μ1,μ2,μ3) = x2 − xy + y2 + μ1(2x + y − 1) + μ2(2 − x − 2y) + μ3( −x − 1)**

**These are NPTEL Data Science for Engineers Assignment 5 Answers**

More Solutions of Data Science for Engineers: Click Here

More Nptel Courses: Click here

**Course Name: Data Science for Engineers**

**Course Link: Click Here**

**These are NPTEL Data Science for Engineers Assignment 5 Answers**

**Q1. Which of the following statements is/are not TRUE with respect to the multi variate optimization?**

I – The gradient of a function at a point is parallel to the contours

II – Gradient points in the direction of greatest increase of the function

III – Negative gradient points in the direction of the greatest decrease of the function

IV – Hessian is a non-symmetric matrix

a. I

b. II and III

c. I and IV

d. III and IV

I – The gradient of a function at a point is parallel to the contours

II – Gradient points in the direction of greatest increase of the function

III – Negative gradient points in the direction of the greatest decrease of the function

IV – Hessian is a non-symmetric matrix

**Answer: c. I and IV**

**Q2. The solution to an unconstrained optimization problem is always the same as the solution to the constrained one.**

a. True

b. False

**Answer: b. False**

**These are NPTEL Data Science for Engineers Assignment 5 Answers**

**Q3. Gradient based algorithm methods compute**

a. only step length at each iteration

b. both direction and step length at each iteration

c. only direction at each iteration

d. none of the above

**Answer: b. both direction and step length at each iteration**

**Q4. For an unconstrained multivariate optimization given f(x¯¯¯), the necessary second order condition for x¯¯¯∗ to be the minimizer of f(x) is**

a. ∇2f(x¯¯¯∗) must be negative definite.

b. ∇2f(x¯¯¯∗) must be positive definite.

c. ∇f(x¯¯¯∗)=0

d. f”(x¯¯¯∗)>0

**Answer: b. ∇2f(x¯¯¯∗) must be positive definite.**

**These are NPTEL Data Science for Engineers Assignment 5 Answers**

**Use the following information to answer Q5, 6, 7 and 8minx1,x2∈R f(x1,x2)=x21+4×22−2×1+8×2.**

**Q5. Which among the following is the stationary point for f(x1,x2)?**

a. (0,0)

b. (1,−1)

c. (−1,−1)

d. (−1,1)

**Answer: b. (1,−1)**

**Q6. Find the eigen values corresponding to Hessian matrix of f.**

a. 1,−1

b. 1,1

c. 2,8

d. 0,2

**Answer: c. 2,8**

**These are NPTEL Data Science for Engineers Assignment 5 Answers**

**Q7. Find the minimum value of f.**

a. 0

b. −5

c. −1

d. 1

**Answer: b. −5**

**Q8. What is the minimum value of f(x1,x2)csubject to the constraint x1+2×2=7?**

a. −5

b. −1

c. 27

d. 0

**Answer: c. 27**

**These are NPTEL Data Science for Engineers Assignment 5 Answers**

**Q9. Find the maximum value of f(x,y)=49−x2−y2 subject to the constraint x+3y=10.**

a. 49

b. 46

c. 59

d. 39

**Answer: d. 39**

**These are NPTEL Data Science for Engineers Assignment 5 Answers**

**Q10. Consider an optimization problem minx1,x2 x2−xy+y2 subject to the constraints 2x+y≤1x+2y≥2x≥−1Find the lagrangian function for the above optimization problem.**

a. L(x,y,µ1,µ2,µ3) = x2 − xy + y2 + µ1(2x + y − 1) + µ2(2 − x − 2y)+µ3( − x − 1)

b. L(x,y,µ1,µ2,µ3) = x2 − xy + y2 + µ1(2x + y − 1) + µ2(x + 2y − 2) + µ3( − x − 1)

c. L(x,y,µ1,µ2,µ3) = x2 − xy + y2 + µ1(2x + y − 1) + µ2(x + 2y − 2) + µ3(x + 1)

d. L(x,y,µ1,µ2,µ3) = x2 − xy + y2 + µ1(1 − 2x − y) + µ2(2 − x − 2y) + µ3( − x − 1)

**Answer: c. L(x,y,µ1,µ2,µ3) = x2 − xy + y2 + µ1(2x + y − 1) + µ2(x + 2y − 2) + µ3(x + 1)**

**These are NPTEL Data Science for Engineers Assignment 5 Answers**

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