# Data Science for Engineers | Week 4

**Session: JULY-DEC 2023**

**Course Name: Data Science for Engineers**

**Course Link: Click Here**

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

#### Q1. Let f(x)=x3+3×2−24x+7. Select the correct options from the following:

x=2 will give the maximum for f(x).

x=2 will give the minimum for f(x).

Maximum value of f(x) is 87.

The stationary points for f(x) are 2 and 4.

**Answer: c, d**

**Q2. Find the gradient of f(x,y)=x2y at (x,y)=(1,3).**

∇f=[16]

∇f=[61]

∇f=[69]

∇f=[33]

**Answer: ∇f=[61]**

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

**Q3. Find the Hessian matrix for f(x,y)=x2y at (x,y)=(1,3).**

∇2f=[32 20]

∇2f=[33 30]

∇2f=[62 20]

∇2f=[63 30]

**Answer: ∇2f=[62 20]**

**Q4. Let f(x,y)=−3×2−6xy−6y2. The point (0,0) is a**

saddle point

maxima

minima

**Answer: saddle point**

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

**Q5. For which numbers b is the matrix A=[1b b9] positive definite?**

−3<b<3

b=3

b=−3

−3≤b≤3

**Answer: −3<b<3**

**Q6. Consider f(x)=x3−12x−5. Which among the following statements are true?**

f(x) is increasing in the interval (−2,2).

f(x) is increasing in the interval (2,∞).

f(x) is decreasing in the interval (−∞,−2).

f(x) is decreasing in the interval (−2,2).

**Answer: b, c**

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

**Q7. Consider the following optimization problem: maxx∈Rf(x), wheref(x)=x4+7×3+5×2−17x+3Let x∗ be the maximizer of f(x). What is the second order sufficient condition for x∗to be the maximizer of the function f(x)?**

4×3+21×2+10x−17=0

12×2+42x+10=0

12×2+42x+10>0

12×2+42x+10<0

**Answer: 12×2+42x+10>0**

**Q8. In optimization problem, the function that we want to optimize is called**

Decision function

Constraints function

Optimal function

Objective function

**Answer: Objective function**

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

**Q9. The optimization problem minx f(x) can also be written as maxx f(x).**

True

False

**Answer: False**

**Q10. Gradient descent algorithm converges to the local minimum.**

True

False

**Answer: False**

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

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**Course Name: Data Science for Engineers**

**Course Link: Click Here**

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

**Q1. Let f(x)=x3+6×2−3x−5. Select the correct options from the following:**

a. −2+√5will give the maximum for f(x).

b. −2+√5will give the minimum for f(x).

c. The stationary points for f(x) are −2+√5 and −2−√5.

d. The stationary points for f(x) are −4 and 0.

**Answer: a, c**

**Use the following information to answer Q2 and Q3.Consider the following optimization problem:maxxϵRf(x), where f(x)=x4+7×3+5×2−17x+3Let x∗be the maximizer of f(x).**

**Q2. What is the second order sufficient condition for x∗ to be the maximizer of the function f(x)?**

a. 4×3+21×2+10x−17=0

b. 12×2+42x+10=0

c. 12×2+42x+10>0

d. 12×2+42x+10<0

**Answer: d. 12×2+42x+10<0**

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

**Q3. Find the value of x∗.**

a. −4.48

b. 0.66

c. −1.43

d. 4.45

**Answer: c. −1.43**

**Q4. Let f(x)=2sinx,0≤x≤2π. Select the correct options from the following:**

a. π2is the global maximum of f(x).

b. π is the global minimum of f(x).

c. 3π2 is the global maximum of f(x).

d. 3π2 is the global minimum of f(x).

**Answer: b, c**

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

**Use the following information to answer Q5, Q6, Q7 and Q8.Let f(x)=2×21+3x1x2+3×22+x1+3×2.**

**Q5. Find the gradient for f(x).**

a. ▽f=[4×1+3×2+13×1+6×2+3]

b. ▽f=[3×1+6×2+34×1+3×2+1]

c. ▽f=[4×1+3x23x1+6×2]

d. ▽f=[4×2+3×1+13×2+6×1+3]

**Answer: a. ▽f=[4×1+3×2+13×1+6×2+3]**

**Q6. Find the stationary point for f(x).**

a. 0.6, 0.4

b. −0.6, −0.4

c. 0.2, −0.6

c. 0.2, 0.6

**Answer: c. 0.2, −0.6**

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

**Q7. Find the Hessian matrix for f(x).**

a. ▽2f=[2336]

b. ▽2f=[3333]

c. ▽2f=[4336]

d. ▽2f=[6334]

**Answer: c. ▽2f=[4336]**

**Q8. The stationary point obtained in Q6 is a**

a. maxima

b. minima

c. saddle point

**Answer: b. minima**

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

**Q9. Let f(x1,x2)=4×21−4x1x2+2×22. Select the correct options from the following:**

a. (2, 4) is a stationary point of f(x).

b. (0, 0) is a stationary point of f(x).

c. The Hessian matrix ▽2f is positive definite.

d. The Hessian matrix ▽2f is not positive definite.

**Answer: b, c**

**Q10. In optimization problem, the function that we want to optimize is called**

a. Decision function

b. Constraints function

c. Optimal function

d. Objective function

**Answer: d. Objective function**

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

**Q11. The optimization problem minxf(x) can also be written as maxxf(x).**

a. True

b. False

**Answer: a. True**

**Q12. In the gradient descent algorithm, the step size should always be same for each iteration.**

a. True

b. False

**Answer: b. False**

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

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