Data Science for Engineers | Week 4

Session: JAN-APR 2024

Course Name: Data Science for Engineers

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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: B, C


Q2. Find the gradient of f(x,y)=x2yat(x,y)=(1,3).
∇f=[1 6]
∇f=[6 1]
∇f=[6 9]
∇f=[3 3]

Answer: ∇f=[1 6]


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These are NPTEL Data Science for Engineers Assignment 4 Answers


Q3. Find the Hessian matrix for f(x,y)=x2yat(x,y)=(1,3).
∇2f=[3 2 2 0]
∇2f=[3 3 3 0]
∇2f=[6 2 2 0]
∇2f=[6 3 3 0]

Answer: ∇2f=[6 2 2 0]


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

Answer: maxima


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These are NPTEL Data Science for Engineers Assignment 4 Answers


Q5. For which numbers b is the matrix A=[1 b b 9] 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, d


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These are NPTEL Data Science for Engineers Assignment 4 Answers


Q7. Consider the following optimization problem:
maxx∈Rf(x), where
f(x)=x4+7×3+5×2−17x+3
Let 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: d. 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


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These are NPTEL Data Science for Engineers Assignment 4 Answers


Q9. The optimization problem minxf(x) can also be written as maxxf(x).
True
False

Answer: False


Q10. Gradient descent algorithm converges to the local minimum.
True
False

Answer: True


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These are NPTEL Data Science for Engineers Assignment 4 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 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: b, c


Q2. Find the gradient of f(x,y)=x2y at (x,y)=(1,3).
∇f=[16]
∇f=[61]
∇f=[69]
∇f=[33]

Answer: ∇f=[ 6 1 ]


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=[ 6 2 2 0 ]


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

Answer: maxima


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), where
f(x)=x4+7×3+5×2−17x+3
Let 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: True


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+3
Let 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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NPTEL Data Science for Engineers Assignment 4 Answers
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