Smallest Possible Slope for Tangent Line to y=2x^3-6x^2+10x+3 on Interval [-2,2]

  • Thread starter Thread starter Emethyst
  • Start date Start date
  • Tags Tags
    Slope
Click For Summary
SUMMARY

The smallest possible slope for a tangent line to the function y=2x^3-6x^2+10x+3 on the interval [-2,2] is determined by analyzing the derivative of the function. The first derivative must be calculated to find critical points, while the second derivative is necessary to identify local minima. The critical point at x=1 yields a slope of 4, which is confirmed through the application of the Algorithm for Extreme Values. This approach effectively identifies the minimum slope within the specified interval.

PREREQUISITES
  • Understanding of calculus concepts, specifically derivatives and critical points
  • Familiarity with the Algorithm for Extreme Values
  • Knowledge of second derivatives and their role in identifying local minima
  • Ability to perform implicit differentiation
NEXT STEPS
  • Study the process of finding critical points using first and second derivatives
  • Learn about the Algorithm for Extreme Values in more depth
  • Practice implicit differentiation techniques with various polynomial functions
  • Explore applications of derivatives in real-world optimization problems
USEFUL FOR

Students studying calculus, particularly those focusing on optimization problems, as well as educators seeking to enhance their teaching methods in derivative applications.

Emethyst
Messages
117
Reaction score
0

Homework Statement


What is the smallest possible slope for a tangent line to y=2x^3-6x^2+10x+3 on the interval [-2,2]?



Homework Equations


implicit differentation



The Attempt at a Solution


Another question that should be easy, yet that I am finding frustrating. To me what it looks like I need to do is take the derivative of the formula and then find any other critical points to use the Algorithm for Extreme Values. The only problem I seem to be having is finding these other critical points because the derivative will not factor. I know 1 is the value for x I need, because the answer is 4 (found the value for x through simple guess and check), but I do not know how to go about finding it or a similar value. Any help for this question would be great, thanks in advance.
 
Physics news on Phys.org
Hi Emethyst!

You want to find the minimum of the derivative (since it is always positive). So you have to look for zeros of the second derivative, not the first.
 
Ohh now I see where the 1 came from, thanks for the help yyat, don't think I would've spotted that otherwise :smile:
 

Similar threads

Can a Function Have Two Different Tangent Lines at the Same Point?
  • · Replies 2 ·
Replies
2
Views
2K
Find the equation of a tangent line to y = x^2?
  • · Replies 11 ·
Replies
11
Views
5K
Can you clarify your understanding for parts (a) and (b)?
  • · Replies 11 ·
Replies
11
Views
2K
Quadratics Having a Common Tangent
  • · Replies 1 ·
Replies
1
Views
1K
Write the tangent lines to the curve
  • · Replies 1 ·
Replies
1
Views
1K
Find the slope of the tangent line to the function's inverse
  • · Replies 13 ·
Replies
13
Views
5K
Finding the Slope of Secant and Tangent Lines
  • · Replies 2 ·
Replies
2
Views
3K
Equations of two Lines that are Tangent to a Parabola
  • · Replies 8 ·
Replies
8
Views
4K
Find eqn for "normal" line to the tangent- folium
  • · Replies 6 ·
Replies
6
Views
2K
Finding a slope at a point on quadratic (intuition of limit)
  • · Replies 4 ·
Replies
4
Views
1K