The document provides instructions and grading guidelines for a mathematical foundations assignment. It asks the student to submit their work by uploading pictures of handwritten calculations along with explanations in a word document. It then lists four problems to solve related to maxima/minima of functions, finding the slope and equation of a line, and identifying critical points. The grading scale ranges from A to F based on whether the assignment was completed and submitted on time.

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Mathematical Foundations
Instructions
Please share your answers filled in-line in the word document. Submit code wherever applicable.
Mathematical calculations which are manually performed should be uploaded with a picture
along with the explanation in a word document.
Please ensure you update all the details:
Name: Ganesh Venkatraman Gaonkar
Batch ID: 18082021
Topic: Mathematical Foundations
Note: Submit pictures of mathematical calculations
Grading Guidelines:
1. An assignment submission is considered complete only when correct and executable code(s) are
submitted along with the documentation explaining the method and results. Failing to submit either
of those will be considered an invalid submission and will not be considered for evaluation.
2. Assignments submitted after the deadline will affect your grades.
Grading:
Ans Date Ans Date
Correct On time A 100
80% & above On time B 85 Correct Late
50% & above On time C 75 80% & above Late
50% & below On time D 65 50% & above Late
E 55 50% & below
Copied/No Submission F 45
● Grade A: (>= 90): When all assignments are submitted on or before the given deadline
● Grade B: (>= 80 and < 90):
o When assignments are submitted on time but less than 80% of problems are completed.
(OR)
o All assignments are submitted after the deadline.

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● Grade C: (>= 70 and < 80):
o When assignments are submitted on time but less than 50% of the problems are
completed.
(OR)
o Less than 80% of problems in the assignments are submitted after the deadline.
● Grade D: (>= 60 and < 70):
o Assignments submitted after the deadline and with 50% or less problems.
● Grade E: (>= 50 and < 60):
o Less than 30% of problems in the assignments are submitted after the deadline.
(OR)
o Less than 30% of problems in the assignments are submitted before the deadline.
● Grade F: (< 50): No submission (or) malpractice.
Problem Statements
Q1) Find the maximum and minimum values of the function: x^3 - 3x^2 - 9x + 12
ANS:
Given statement is
F(x)= x3
-3x2
-9x+12
On differentiating both sides,w.r.tx,we get
F(x) = 3x2
– 6x – 9 -------(1)
Therefore d/dx xn
= nxn-1
For maxima or minima,
Put f(x)=0
3x2 – 6x – 9 =0
x2
- 2x - 3=0
x2
– 3x + x – 3 = 0
X(x-3) + 1(x- 3) = 0
(x-3) (x+1) =0
x=3 or x= -1
When x = 3 , we get

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f(3) = 6x3-9=19-9=9
f(3) 0
f(x) is minimum at x =3
and
minimum value = f(3)
= (3)3
– 3(3)2
– 9*3+12
= 27-27-27+12
= -15
Hence maximum value of f (x) is -15
When x = -1, we get
f (-1) = 6*(-1)-9 = - 6 – 9 = -15
f (-1)
f (x) is maximum at x= -1
and
maximum value = f (-1)
= (-1)3 – 3(-1)2 – 9(-1) + 12
= -1 – 3 + 9 + 12
=17
Hence maximum value of f(x) = 17
Q2) Calculate the slope and the equation of a line which passes through the points (-1, -1) (3, 8)
Ans: Slope = y2 – y1 / x2 – x1
= 8+1 / 3+1
= 9/4
= 2.25

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Q3) Solve for w’(z) when
ANS:
W(z) = 4z – 5 – 2 -z
4z – z – 5 -2 =0
3z – 7 = 0
z = 7/3
Therefore z = 7/3
Q3) Consider Y(x)= 2x^3+6x^2+3x. Identify the critical values and verify if it is the maxima or
minima.
Ans:
Critical points:
Dy/dx = 6x2 + 12x + 3 = 0
2x2 + 4x + 1 = 0
X= - 1 -1/ ,-1+1/
Maxima or Minima:
At x = -1 – 1/
= d2y/dx2
= 12 (-1/ )
Y (x) has maximum value at x = -1-1/
As well as min Value is Y (-1+1/ )
Q4) Determine the critical points and obtain relative minima or maxima of a function defined by