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), 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: 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+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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