Nptel Data Science for Engineers Assignment 4 Answers

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Nptel Data Science for Engineers Assignment 4 Answers
Nptel Data Science for Engineers Assignment 4 Answers

Nptel Data Science for Engineers Assignment 4 Answers (July-Dec 2024)


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

−2+5–√will give the maximum for f(x)

−2+5–√will give the minimum for f(x)

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

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

Answer: The stationary points for f(x) are −4 and 0


Q2What 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


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Q3.Find the value of x∗.
−4.48
0.66
−1.43
4.45

Answer: −1.43


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

π2 is the global maximum of f(x).

π is the global minimum of f(x).

3π2 is the global maximum of f(x).

3π2 is the global minimum of f(x)

Answer: π2 is the global maximum of f(x)., 3π2 is the global minimum of f(x)


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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)


Q6.Find the stationary point for f(x1,x2).
0.6, 0.4
−0.6, −0.4
0.2, −0.6
0.2, 0.6

Answer: 0.2, −0.6


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Q7.Find the Hessian matrix for f(x1,x2)
.

a) ∇2f=[2336]

b) ∇2f=[3333]

c) ∇2f=[4336]

d) ∇2f=[6334]

Answer: c) ∇2f=[4336]


Q8. The stationary point obtained in the previous question is
maxima
minima
saddle point

Answer: minima


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

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

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

The Hessian matrix ∇2f is positive definite.

The Hessian matrix ∇2f is not positive definite.

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

The Hessian matrix ∇2f is positive definite.


Q10. In optimization problem, the function that we want to optimize is called
Decision function
Constraints function
Optimal function
Objective function

See also  Nptel Data Science for Engineers Week 2 Assignment Answers

Answer: Objective function


Q11.The optimization problem minxf(x) can also be written as maxxf(X)
True
False

Answer: False


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

Answer: False


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Nptel Data Science for Engineers Assignment 4 Answers (JAN-APR 2024)

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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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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

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Answer: False


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

Answer: True


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Nptel Data Science for Engineers Assignment 4 Answers (JULY-DEC 2023)

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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 ]


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,


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


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



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Nptel Data Science for Engineers Assignment 4 Answers (JAN-APR 2023)

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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).

See also  Nptel Data Science for Engineers Assignment 3 Answers

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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