Question

The minimum value of the slope of the curve

Answer 1

\[y=x^5+x^3-2x\]

Answer 2

this means you need the minimum value of the second derivative....
find the second derivative and solve when setting it equal to zero....

Answer 3

\[y''=20x^3+6x\] how can I get the value there are 3 solutions?

Answer 4

If they happen to all be the same solution, then that's the min.

If the solutions are unique, so you have 3, then you have 3 local minimums, but one of the 3 should be the absolute minimum when you find the slopes at each of those 3 points.

At least I think that's true... @dpaInc might be able to confirm or clarify.

Answer 5

your second derivative is correct.... and
\(\Large 20x^3+6x=0 \rightarrow x(20x^2+6)=0 \)
notice that second factor (20x^2+6) cannot equal zero...

Answer 6

I got it so the only value is x=0? is it the min. slope already?

Answer 7

the minimum slope OCCURS at x=0

Answer 8

thanks! can I ask why is the second derivative? is'nt it is the point of inflection?

Answer 9

yes... it's the point of inflection... but think of it this way....
when you set the first derivative equal to zero, you're looking for the max/min of the FUNCTION.
when you set the second derivative equal to zero, you're looking for the max/min of the FIRST DERIVATIVE.

Answer 10

and isn't that what you wanted? the minimum value of the slope of the function?

Answer 11

and the max/min of the FIRST derivative is the point of inflection on the function.

Answer 12

Thanks a lot!

Answer 13

Great explanation... thanks!