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

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In summary, to find the smallest possible slope for a tangent line to y=2x^3-6x^2+10x+3 on the interval [-2,2], we can use implicit differentiation and find the zeros of the second derivative.
  • #1
Emethyst
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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.
 
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  • #2
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.
 
  • #3
Ohh now I see where the 1 came from, thanks for the help yyat, don't think I would've spotted that otherwise :smile:
 

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

What does "smallest possible slope" mean?

The term "smallest possible slope" refers to the smallest angle or degree of inclination of a line or surface. It describes the rate of change between two points on a graph or surface.

How is the smallest possible slope calculated?

The smallest possible slope is calculated by dividing the change in the dependent variable (y) by the change in the independent variable (x) between two points on a line or surface. This can be represented by the equation: slope = (y2 - y1) / (x2 - x1).

What is the significance of the smallest possible slope?

The smallest possible slope is significant because it represents the steepest rate of change between two points. It is also used to determine the direction of a line or surface, as well as the strength of the relationship between two variables.

How can the smallest possible slope be used in real-world applications?

The smallest possible slope is used in various fields, such as engineering, physics, and economics, to analyze and predict changes in data. For example, it can be used to determine the speed of an object, the slope of a hill, or the rate of growth of a population.

Can the smallest possible slope be negative?

Yes, the smallest possible slope can be negative. A negative slope indicates a downward or decreasing trend, while a positive slope indicates an upward or increasing trend. The magnitude of the slope is more important than the sign when interpreting its meaning.

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