# Data Science for Engineers | Week 4

### Session: JULY-DEC 2023

Course Name: Data Science for Engineers

#### 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.

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

#### 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]

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

#### 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

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).

#### 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

Q8. In optimization problem, the function that we want to optimize is called
Decision function
Constraints function
Optimal function
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

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

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

Course Name: Data Science for Engineers

#### 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.

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

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

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).

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]

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

These are NPTEL Data Science for Engineers Assignment 4 Answers

Q7. Find the Hessian matrix for f(x).
a. ▽2f=
b. ▽2f=
c. ▽2f=
d. ▽2f=

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

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

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

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