Explain why the function y = xe^x have a point of inflection? explain fully.?

Question

Explain why the function y = xe^x have a point of inflection? explain fully.

Answer 1

A function can be visually represented by its graph, and the 1st derivative of the function is associated with the slope of the tangential line touching the graph.

The 2nd derivative of a function has to do with the rate at which the slope of the tangential line increases or decreases. In other words, y'' gives you a clue of how the graph bends. If y'' is positive, it means that the slope of the tangential line is increasing, which implies that the graph has a U-shaped look. If y'' is negative, it means that the graph has a reversed U-shaped look. The inflection point is where the graph changes the bending from a U-shaped to a reversed U-shaped look.

1st derivative of y,

y' = xe^x+ e^x

2nd derivative of y,

y'' = xe^x+ e^x+ e^x = (x+ 2)e^x

There is an inflection point where y'' equals zero,

y'' = 0
(x+ 2)e^x = 0

e^x can't be zero for any value of x, so

x+ 2 = 0
x = -2 <--- place where there is an inflection point

Answer 2

the pt of inflection (poi) is the point at which the function changes the direction of the "rate of change" ... more clearly, if a function changes a convexity to a concavity or vice versa at some point, then this point is called an inflecion point

by definition, the second derivative must be zero at that point...ie. no change in rate....

we can find that this function has x=-2 as the poi...

notice how its values change at points x=-3, -2.5, -2.25, -2, -1.75, -1.5, -1.25... find the differences in values...and see how they progress.... that will explain why it has a poi at x=-2

Answer 3

the 1st spinoff provides you the slope of the curve at that factor. that's why we set it to 0 to locate turning factors. the 2d spinoff provides you the cost of replace of the slope. If the 2d spinoff is >0, the curve is concave up (like a U shape), so we've an area minimum. If the 2d spinoff is < 0, that's concave down (like an the other way up U) and there is an area optimum. while the 2d spinoff is 0, the curve is changing its concavity (like the midsection of an S) and it extremely is talked approximately as a table sure factor of horizontal factor of inflection.

Answer 4

it doesn't have a point of inflection whatever variable you use it will always increase....