Maxima, minima, and the mvt application

Click For Summary
SUMMARY

The discussion centers on applying the Mean Value Theorem (MVT) to determine the interval over which the function f(x) is increasing, given the conditions f''(x) ≥ -1 and f'(1) = 3. The conclusion drawn is that since f'(x) ≥ -x + 4, the function is definitely increasing for x values less than 4. This analysis utilizes the relationship between the second derivative and the first derivative to establish the behavior of the function.

PREREQUISITES
  • Understanding of the Mean Value Theorem (MVT)
  • Knowledge of derivatives, specifically first and second derivatives
  • Familiarity with inequalities involving functions
  • Basic calculus concepts, including intervals of increase and decrease
NEXT STEPS
  • Study the Mean Value Theorem and its applications in calculus
  • Learn how to analyze the behavior of functions using first and second derivatives
  • Explore inequalities in calculus and their implications on function behavior
  • Practice problems involving intervals of increase and decrease for various functions
USEFUL FOR

Students preparing for calculus exams, educators teaching derivative concepts, and anyone looking to strengthen their understanding of function behavior through calculus principles.

T Botha
Messages
1
Reaction score
0
Hi there

I'm prepping for a big test tomorrow and I'm really struggling with this question:If f′′(x)≥−1, x belongs to (−15,15), and f′(1)=3, find the interval over which x is definitely increasing.I'm struggling with substitution because I just don't seem to have enough values. Is there a formula that gives an answer? Please let me know.

:( this is making me so sad.
 
Last edited:
Physics news on Phys.org
Since f''(x)\ge -1, f'(x)\ge -x+ C. Since f'(1)= 3, we know that C can be as large as 4. If f'(x)\ge -x+ 4 it will be positive for x less than 4.
 

Similar threads

Undergrad How to find local maxima and minimas of undefined functions?
  • · Replies 2 ·
Replies
2
Views
3K
Finding the Local Max/Min of f(x,y) on C
  • · Replies 14 ·
Replies
14
Views
2K
What are the local maxima and minima of F(x)=(x^2)/(x+1)?
  • · Replies 3 ·
Replies
3
Views
2K
Absolute vs. Relative Maxima and Minima
  • · Replies 2 ·
Replies
2
Views
4K
Identifying Critical Points in e^x(1-cosy): Local Extrema or Saddles?
  • · Replies 9 ·
Replies
9
Views
3K
A few questions to make sure I understand Fourier series
  • · Replies 3 ·
Replies
3
Views
2K
MHB Gibbs Phenomenon:The maxima get smaller,the minima get bigger
  • · Replies 1 ·
Replies
1
Views
2K
Find all critical points, and identify them as minima, maxima, or
  • · Replies 5 ·
Replies
5
Views
2K
MHB Use Variables to Create Function
  • · Replies 6 ·
Replies
6
Views
2K
MHB Evaluate Inverse of Hi M.H.B.: Math Problem
  • · Replies 3 ·
Replies
3
Views
3K