SOLUTION: For each x>0, let m(x) be the slope of the line which goes through the point (0,0) and the point (x,y) on the curve y=(x^2)(e^(-0.25x)). What is the largest possible value for the

Question 1052798: For each x>0, let m(x) be the slope of the line which goes through the point (0,0) and the point (x,y) on the curve y=(x^2)(e^(-0.25x)). What is the largest possible value for the m(x)?
note: it is e raised to (-0.25x)
Work: I believe I've taken the correct first derivative and second derivative of the curve using chain rule and product rule, But I don't know what the question is asking for or how to answer it. y=(x^2)(e^(-0.25x)) -> 1st derivative y'=2xe^((-1/4)x) + x^2((-1/4)(e^(-1/4)x)) -> 2nd derivative:
y''= 2(e^((-1/4)x) + 2x(e^((-1/4)x)(-1/4) + 2x((-1/4)e^((-1/4)x) + x^2((-1/16)(e^(-1/4)(x)))
If possible please check my derivatives, thanks in advance.
Answer by Alan3354(69443) (Show Source): You can put this solution on YOUR website!
For each x>0, let m(x) be the slope of the line which goes through the point (0,0) and the point (x,y) on the curve y=(x^2)(e^(-0.25x)). What is the largest possible value for the m(x)?
-------------
y=
y' =
---
y" =
y" =
y" =
---
Set y" = 0

---
Ignore
---

Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)

Quadratic equation (in our case ) has the following solutons:

For these solutions to exist, the discriminant should not be a negative number.

First, we need to compute the discriminant : .

Discriminant d=128 is greater than zero. That means that there are two solutions: .

Quadratic expression can be factored:

Again, the answer is: 13.6568542494924, 2.34314575050762. Here's your graph:

---

RELATED QUESTIONS
For each x>0, let m(x) be the slope of the line which goes through the point (0,0) and... (answered by ikleyn)

(2 points) Write an equation for the line through the points (8,10) and (4,3) in... (answered by josgarithmetic)

A curve is given: y = e^-x^2. Find the point on the curve where the slope of the tangent... (answered by Fombitz)

Find equations of the tangent line and normal line at the curve at the given point. a). (answered by Fombitz)

Identify each expression that represents the slope of a tangent to the curve y=1/x+1 at... (answered by ikleyn)

y+4=-4(x-2) What is the equation in standard form of a parallel line that passes... (answered by stanbon)

please help me fine the point slope form and slope intercept passing through (-2, -5) (6... (answered by checkley71)

Find the equation of the tangent to the curve y = e^x at the point where it crosses the... (answered by ewatrrr)

Let A be a point on the curve C: x^2 + y^2 - 2x - 4 = 0. If the tangent line to C at A... (answered by josgarithmetic)