Find the x coordinate of the stationary point of the following curves

In summary, the first problem asks us to find dy/dx and the exact x coordinate of the stationary point for the function y=(4x^2+1)^5. The equation for dy/dx is 40x(4x^2+1)^4=0 and to find the x coordinate, we can solve the equations 40x=0 and (4x^2+1)^4=0. The second problem involves finding dy/dx and the exact x coordinate of the stationary point for the function y=x^2/lnx. The equation for dy/dx is (2xlnx-x)/(lnx)^2=0 and to find the x coordinate, we can solve the equation x(
  • #1
studentxlol
40
0

Homework Statement



Find dy/dx and determine the exact x coordinate of the stationary point for:

(a) y=(4x^2+1)^5

(b) y=x^2/lnx

Homework Equations




The Attempt at a Solution



(a) y=(4x^2+1)^5

dy/dx=40x(4x^2+1)^4

40x(4x^2+1)^4=0

Find x... How?

(b) y=x^2/lnx

dy/dx=2xlnx-x^2 1/x / (lnx)^2

2xlnx-x^2 1/x / (lnx)^2=0

Find x... How?
 
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  • #2
studentxlol said:
40x(4x^2+1)^4=0

Find x... How?

re 1st prob:

Then either 40x = 0 or (4x^2+1)^4 = 0.

and solve the above two equations.
 
  • #3
You are aware that [itex]x^2/x= x[/itex] aren't you?

[itex]y= x^2/ln(x)[/itex]: [itex]y'= (2xln(x)- x)/(ln(x))^2= 0[/itex]
Use parentheses! What you wrote was [itex]y'= 2x ln(x)- (x/(ln(x))^2)= 0[/itex].

Multiply both sides of the equation by [itex](ln(x))^2[/itex]
and you are left with 2x ln(x)- x= x(2ln(x)- 1)= 0. Can you solve that?
 

1. What is a stationary point in a curve?

A stationary point is a point on a curve where the gradient (slope) is zero. This means that at that point, the curve is neither increasing nor decreasing in value.

2. How do you find the x-coordinate of a stationary point?

To find the x-coordinate of a stationary point, you need to take the derivative of the curve and set it equal to zero. Then, solve for x to find the x-coordinate of the stationary point.

3. What is the significance of finding the x-coordinate of a stationary point?

The x-coordinate of a stationary point can tell us important information about the behavior of the curve. For example, it can indicate the maximum or minimum values of the curve, or points of inflection where the curve changes from concave up to concave down.

4. Can there be more than one stationary point on a curve?

Yes, there can be multiple stationary points on a curve. A curve may have a local maximum or minimum at one point, and a global maximum or minimum at another point. It can also have points of inflection where the curve changes direction.

5. Are all stationary points of a curve visible on a graph?

No, not all stationary points are visible on a graph. Some may occur in areas where the curve is very steep, making it difficult to see the point where the gradient is zero. In these cases, we can use mathematical methods to find the coordinates of the stationary points.

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