Optimization: Let f(x) = 3x^5 + 5x^4 + 9x + 10....

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whiteti

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Let f(x)=3x5+5x4+9x+10 . Find the equation of the tangent line to f that has the smallest slope.

Do I start by taking the derivative of the function?

If so, Do I use Taylor expansion to find the equation or?
 
whiteti said:
Let f(x)=3x5+5x4+9x+10 . Find the equation of the tangent line to f that has the smallest slope.

Do I start by taking the derivative of the function?

If so, Do I use Taylor expansion to find the equation or?
Click to expand...

Yes - you have to find derivative - but that is a small part of the problem.

However, you do not need to "touch" Taylor series expansion - for this problem!!

You have to start by thinking - what is being asked for in this problem.

Find the equation of the tangent line to f that has the smallest slope

So first you have to figure out:

What is the equation of a line?

What is the slope of a line - in that equation?

What is the slope of the tangent line to a curve?

How do you find the smallest slope?

Start answering these questions ....
 
y-y1=m(x-x1) is the equation of the tangent line

not sure how to get there
 
whiteti said:
y-y1=m(x-x1) is the equation of the tangent line

not sure how to get there
Click to expand...

Good - that would be the "form" of your final answer.

In the equation above - what is the slope of the line?
 
whiteti said:
F'(x)=15x4+20x3+9 Correct
Click to expand...
Do you understand that the first derivative of a function is the slope of the function and thus the slope of the line tangent to the function?

Do you understand that the first derivative of a function is itself a function?

How do you find local extrema of any differentiable function?

Is the first derivative of this function a differentiable function?

So how do you find its minimum and why would you want to?
 
JeffM said:
Do you understand that the first derivative of a function is the slope of the function and thus the slope of the line tangent to the function?

Do you understand that the first derivative of a function is itself a function?

How do you find local extrema of any differentiable function?

Is the first derivative of this function a differentiable function?

So how do you find its minimum and why would you want to?
Click to expand...

Do I need to find local extrema?
 
whiteti said:
Do I need to find local extrema?
Click to expand...
The problem asks you to find the smallest slope. Does that not indicate to you a minimum? An extremum is either a minimum or a maximum. 90% of the mechanics of differential calculus is about finding a minimum or a maximum, in other words an extremum, of a differentiable function. The thinking comes in figuring what is the relevant function and whether the relevant extremum is a minimum or a maximum.
 
whiteti said:
Do I need to find local extrema?
Click to expand...

If you're talking about local extrema of f`, then, yes, that would work because you're looking for the smallest value of f`(x).

That value is called "the absolute minimum" of f`.

This absolute minimum value of the derivative is also the slope of the tangent-line equation that you seek.
 
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