So far I gained to calculating the derivative 5x^4 +2" and second derivative 20x^3"" yet then I got stuck here. Many thanks ahead of time because that the aid :D
The an initial derivative is offered by
#f"(x) = 5x^4 +2#
Recall that horizontal tangents will occur when #f"(x) = 0#.
#0 = 5x^4 +2#
#2/5 = x^4#
#x= root(4)(2/5)#
This is undefined, as such the role has no horizontal tangents.
To find the tangent with the minimum slope, we seek to minimization #5x^4 + 2#. You"re right that the derivative that this will be #20x^3#. This will have a crucial point in ~ #x= 0#, which will certainly be a minimum since #f""(1) = # and also #f""(1)= +# and also #f""(0) = 0#.
Therefore, the value of #x# that offers the the smallest tangent slope will be #x =0# and also the slope will certainly be #2#.
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Hopefully this helps!
Answer attach
LM
Feb 17, 2018
#5x^4+2>=2#smallest slope possible: #2#
Explanation:
the gradient of a horizontal heat is #0#.
the derivative of a role can be supplied to find the gradient of a line tangent come the graph.
you have offered the derivative that the duty #y = x^5+2x#; that is #5x^4+2#.
all genuine numbers have squares that are either positive, or #0#.
#x^4 = (x^2)^2#
the square of any kind of positive number is likewise positive.the square that #0# is #0#.
this way that #x^4# must constantly be #0# or above.
#5x^4 + 2# must thus be #2# or above.
since no tangent to the graph #y=x^5+2x# can have a gradient equal to #0#, there deserve to be no horizontal tangents.

the smallest slope feasible can be discovered by calculating the value of #x# when the second derivative is #0#.
(note the all gradients #5x^4 +2#, for any kind of real worth of #x#, room nonnegative.)
the second derivative offered here is #20x^3#.
#20x^3 = 0#
#x^3 = 0#
#x = root3(0) = 0#
then the worth of #x# that makes the second derivative #0# deserve to be substituted into the equation because that the first derivative.
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#x = 0#
#5x^4 + 2 = (5*0^4) + 2#
#= (5*0) + 2#
#= 0 + 2#
#= 2#
the the smallest value feasible for the derivative, and also therefore the smallest feasible value because that the slope of the tangent, is #2#.