Finding Maxima & Minima (Practice Questions Set 2)

Duration: 7 min

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Module outline

  1. General Intelligence & Reasoning: Analogy, Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning), Series (Number and Letter Series) (Numerical Relations and Reasoning), Coding Decoding, Blood Relations, Directions (Direction Test), Ranking, Order and Sequence, Logical Sequence Of Words, Syllogisms, Venn Diagram, Deductive and Inductive Reasoning (Logical Deduction and Induction) (Prepositional Reasoning), Selection Decision Table (Decision Making), Data Sufficiency, Puzzle Solving (Puzzles), Data Arrangement, Cubes & Dices, Calendar, Clock, Arithmetic, Number System, HCF LCM, Verbal Analogy (Word Based Analogy), Visual Recognition and Figure Completion, Paper Folding and Cutting, Transformation of Shapes, Clocks and Calendars, Direction Sense, Cubes and Dice, Patterns in Two-Dimensional Figures, Seating Arrangements, Analogy and Classification, Logical Deduction and Induction, Alphabet and Word Tests, Mathematical Operations, Series Completion, Simplification and Approximation, Coding and Decoding, Numerical Reasoning, Assembling and Grouping of Shapes, Three-Dimensional Patterns and Visualization, Vocabulary, Number System, Averages, Mixtures and Alligation, Simple and Compound Interest, Percentages, Algebra, Speed, Time and Distance, Seating and Arrangement Puzzles, Traverse Surveying, Order, Ranking and Sequence, Constraint Puzzles, Syllogisms and Venn Diagrams, Reading Comprehension, Critical Reasoning, Permutations and Combinations, Narrative Sequencing, Project Monitoring, Resources and Safety, Profit, Loss and Discount, Ratio and Proportion, Time and Work, Mensuration, Data Interpretation, Powers, Surds and Logarithms, Sequences and Series, Articles and Determiners, Sentence Completion, Data Sufficiency, Blood Relations, Simple Interest and Compound Interest, Mixture and Alligation, Sequence and Series (Series), Speed Time and Distance, Algebra, Profit and Loss, Powers and Exponents (Surds and Indices), Permutation and Combination
  2. General Awareness: Physics, Chemistry, Biology, History, Geography, Polity, Indian Economy, Environment, Static General Knowledge, GK- Current Affairs, Artificial Intelligence, MS Office Applications, Terms & Abbreviations, Keyboard Shortcuts, Evolution of Indian Banking (Banking & Financial Awareness), Emerging Tech Trends
  3. Engineering Mathematics: Set Theory, Relations, Functions, Graph Theory, Group Theory, Propositional and Predicate Logic, Linear Algebra, Calculus, Probability, Statistics, Signal Space and Optimum Detection, Probability, Statistics
  4. Digital Logic: Digital Systems & Boolean Basics, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Combinational Circuit, Sequential Circuits, Number System, Number Representation, Floating Point Rep, Number Systems and Codes, Registers and Counters, Boolean Algebra and Logic Gates, Microprocessors and Microcontrollers, Logic Families and CMOS Logic, Microprocessors, Logic Gates and Boolean Algebra, Minimization of Boolean Functions, Timing, Hazards and Testing, Logic Families, Combinational Logic Design, Arithmetic Circuits, Waveform Generators and Timer Circuits, Semiconductor Memories and Programmable Logic Devices, Combinational Logic Circuits, Latches and Flip-Flops, Finite State Machines, Sequential Circuits and State Machines, Minimization of Boolean Functions, Arithmetic Circuits, Latches and Flip-Flops, Number Systems and Codes, Registers and Counters
  5. Computer Organisation & Architecture: Basics of COA, Cache Memory Organization, Input Output Organisation, Pipelining, Instr Formats & Modes, Control Unit Design, Programming the Basic Computer, Register Transfer and Microoperations, Semiconductor Memories and Programmable Logic, Processor Organization and Instructions, Memory Hierarchy and Input/Output, Registers and Counters
  6. Programming & Data Structures: C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums, DMA, Macros, Scoping & File Handling, Introduction to DS, Array, Stack, Queue, Linked List, Tree, Graphs, Hashing, HTML Basics, OOPS with C++, Java, Object Oriented Technology
  7. Databases: Basics of DBMS, ER Diagram, Relational Model & Functional Dependencies, Keys & Integrity Constraints, Normalization (1NF - BCNF), Decomposition Properties & 4NF, File Organization & Indexing, Relational Algebra, SQL, Relational Calculus, Transaction Management, Concurrency Control, Database Recovery, Database Security & Authorization
  8. Algorithms: Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Dynamic Programming, Minimum Spanning Trees, Shortest Path Algos
  9. Theory of Computation: Introduction to TOC, Deterministic FA (DFA), Non-Deterministic FA, Regular Expressions, Grammar, Regular Language Properties, Moore & Mealy Machines, Pushdown Automata & CFG, Turing Machines, Complexity Theory
  10. Compiler Design: Intro to Compilers, Lexical Analysis, Grammar & CFG, Syntax Analysis: Top-Down, Syntax Analysis: Bottom-Up, Semantic Analysis & SDT, Intermediate Code Gen, Code Optimization, Run Time Environment
  11. Operating System: Introduction to OS, Process Management, CPU Scheduling, Process Synchronization, Threads & Process Creation, Deadlock, Memory Management, Virtual Memory, Disc Scheduling, File Management, Security, Windows OS, Distributed Systems, Unix OS, Linux OS
  12. Computer Networks: Introduction to CN, DLL: Access Control, DLL: Flow Control, DLL: Error Control, DLL: Framing, Data Link Layer - Ethernet, Net Layer: IPv4 & Proto, Net Layer: IP Addressing, Net Layer:Routing Protocol, Transport Layer Services, TL: Congestion & UDP, Application Layer, Hardware Basics, Data Communication, Website, Browser & Services, Web Design & Development, Network Security, Cyber Threats and Malware Attacks, Cyber Safety and Security
  13. Full Length Mock Tests:
  14. Sectional Tests:
  15. Paper 2 - Full-length Mock Tests:
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This educational video features a calculus lecture by Yash Jain Sir from Knowledge Gate, focusing on finding local maxima and minima using derivatives. The lesson begins with a polynomial function example, demonstrating the standard procedure of finding critical points where the first derivative is zero and classifying them using the second derivative test. The instructor then transitions to a trigonometric function to illustrate a case where multiple stationary points exist, highlighting the periodic nature of trigonometric equations and the resulting infinite solutions.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The video opens with a title card displaying the word 'CALCULUS' amidst various mathematical formulas like integrals and series. The scene transitions to Yash Jain Sir, an educator from Knowledge Gate, standing before a whiteboard. He introduces the problem by writing the function $f(x) = rac{x^3}{3} - x$ in black ink. He states the goal is to find the local maxima and minima for this specific cubic function. He begins the solution process by preparing to differentiate the function with respect to $x$.

  2. 2:00 – 5:00 02:00-05:00

    The instructor proceeds to differentiate the function, writing $f'(x) = x^2 - 1$ on the board. He sets the derivative equal to zero, $f'(x) = 0$, to locate the stationary points (SP). Solving $x^2 - 1 = 0$ gives the critical values $x = 1$ and $x = -1$, which he circles. He then computes the second derivative, $f''(x) = 2x$. By substituting $x=1$, he finds $f''(1) = 2$, which is greater than zero, indicating a local minimum. Conversely, for $x=-1$, $f''(-1) = -2$, which is less than zero, indicating a local maximum. He calculates the corresponding function values, writing $-2/3$ for the minimum and $2/3$ for the maximum.

  3. 5:00 – 7:13 05:00-07:13

    A new example is introduced: $f(x) = 2 \cos x - x$. The instructor differentiates this to obtain $f'(x) = -2 \sin x - 1$. Setting the derivative to zero results in the trigonometric equation $\sin x = -1/2$. He draws a rough graph of the function, showing a wave-like pattern superimposed on a linear decline. He circles the equation $\sin x = -1/2$ and explains that this equation has 'many values of x' due to the periodic nature of the sine function. He notes that unlike the previous polynomial example, there are infinite stationary points, making it impossible to list them all.

The lecture effectively demonstrates the application of the first and second derivative tests for optimization. It contrasts a simple polynomial case with a unique solution set against a trigonometric case with an infinite solution set, reinforcing the importance of analyzing the nature of the function when solving for critical points.

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