Notes from Lecture on Solutions to Exercise 6.4 (Questions 14-18)

Introduction

• Lecture by Dr. Rajesh Singh.
• Focus: Solutions for Exercise from Section 6.4 of Barton and Sherbert.
• Previous questions (1-13) were discussed in earlier videos.
• This session covers questions 14 to 18.

Question 14: Relative Extrema at x = 0

• Objective: Determine if x = 0 is a relative point of extrema for the given functions.
• Theorem Used: Relative Extrema Theorem - find derivatives until the first non-zero derivative.

Part A: Function f(x) = x³ + 2

• First derivative: f'(x) = 3x², f'(0) = 0
• Second derivative: f''(x) = 6x, f''(0) = 0
• Third derivative: f'''(x) = 6, f'''(0) = 6 (non-zero)
• Conclusion: No relative extrema at x = 0 (odd derivative).

Part B: Function g(x) = sin(x) - x

• First derivative: g'(x) = cos(x) - 1, g'(0) = 0
• Second derivative: g''(x) = -sin(x), g''(0) = 0
• Third derivative: g'''(x) = -cos(x), g'''(0) = -1 (non-zero)
• Conclusion: No relative extrema at x = 0 (odd derivative).

Part C: Function h(x) = sin(x) + (1/6)x³

• First derivative: h'(x) = cos(x) + (1/2)x², h'(0) = 1 (non-zero)
• Conclusion: No relative extrema at x = 0.

Part D: Function k(x) = cos(x) - 1 + (1/2)x²

• First derivative: k'(x) = -sin(x) + x, k'(0) = 0
• Second derivative: k''(x) = -cos(x) + 1, k''(0) = 0
• Third derivative: k'''(x) = sin(x), k'''(0) = 0
• Fourth derivative: k⁽⁴⁾(x) = cos(x), k⁽⁴⁾(0) = 1 (non-zero)
• Conclusion: Relative minimum at x = 0 (even derivative).

Question 15: Continuous Function on Closed Interval [a,b]

• Objective: Prove the existence of a point where the second derivative is zero.
• Given: Function f is continuous, and its second derivative exists on (a, b).
• Graph of f intersects line segment joining (a, f(a)) and (b, f(b)) at a point (x₀, f(x₀)).

Application of Mean Value Theorem

• Apply Mean Value Theorem twice for intervals [a, x₀] and [x₀, b].
• Existence of c₁ in [a, x₀] and c₂ in [x₀, b] such that:
  • f'(c₁) = (f(x₀) - f(a)) / (x₀ - a)
  • f'(c₂) = (f(b) - f(x₀)) / (x₀ - b)
• By Rolle's Theorem, there exists c in (c₁, c₂) such that f''(c) = 0.

Question 16: Limit Definition of Second Derivative

• Objective: Derive the formula for the second derivative using limits.
• Start with the limit definition of the derivative to derive the formula for f''(a).
• Provide an example where the limit exists but the second derivative does not.

Example: f(x) = x² sin(1/x) (x ≠ 0), f(0) = 0

• Show existence of limit:
  • f'(x) at x = 0 using limit definition results in 0.
• Show f''(0) does not exist due to lack of continuity in f'.

Question 17: Behavior of Graph with Positive Second Derivative

• Given: A continuous function f on an interval I with f''(x) > 0 on I.
• Objective: Show that f(x) is never below the tangent line at any point c in I.

Application of Taylor's Theorem

• Use Taylor's theorem to express f(x) around c.
• Show that since f''(c) > 0, the graph of f is above the tangent line at c.

Question 18: Taylor's Theorem Application

• Given two functions f and g on interval I with multiple derivatives equal to zero at c.
• Use Taylor's theorem to show the limits of f and g as x approaches c.
• Ensure the conditions for g(c) not equal to 0 to find the limit.

Conclusion

• Closure of the lecture with a reminder to subscribe for updates and further learning resources.