Nptel Data Science for Engineers Assignment 4 Answers
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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
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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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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
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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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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