SOLUTION: At what values of x on the curve y=5+40x^3-3x^5 does the tangent line have the largest slope?

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Question 1180886: At what values of x on the curve y=5+40x^3-3x^5 does the tangent line have the largest slope?
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

At what values of x on the curve
y=5%2B40x%5E3-3x%5E5 does the tangent line have the largest slope?
%28dy%2Fdx%29%285%2B40x%5E3-3x%5E5%29=3%2A40x%5E2-5%2A3x%5E4
%28dy%2Fdx%29%285%2B40x%5E3-3x%5E5%29=120x%5E2-15x%5E4

%28d%5E2y%2Fdx%5E2%29%285%2B40x%5E3-3x%5E5%29=240x-60x%5E3
To find where the slope of the tangent line is the largest, we need to find the inflection points which occur when the second derivative equals zero:
240x-60x%5E3=0
60x%284-x%5E2%29=0
solutions:
60x=0=>x=0
4-x%5E2=0=>%282-x%29%282%2Bx%29=0
%282-x%29=0=>x=2
2%2Bx=0=>x=-2

The roots to the above equation are
x=0, x=-2,x=2
however, we must check each one to see what the slope is at these points.

At x=0, we have
%28dy%2Fdx%29=120%2A0%5E2-15%2A0%5E4=0
.
At x=-2, we have
%28dy%2Fdx%29=120%2A%28-2%29%5E2-15%2A%28-2%29%5E4=240
At x=2, we have
%28dy%2Fdx%29=120%2A%282%29%5E2-15%2A%282%29%5E4=240
Therefore, the tangent line to the curve has the largest slope at
x=2 and x=-2
and the value of the largest slope is 240 at these points.