I need help sketching the graph of 2sqrt(x)-x using differentiation
I don't see why you need to use differentiation to plot the graph of that or any function
I need to find the critical points and absolute minimum and maximum
To sketch the graph of the function 2√(x) - x using differentiation, we will follow these steps:
Step 1: Find the first derivative of the function.
Step 2: Find the critical points by setting the derivative equal to 0.
Step 3: Determine the nature of the critical points by analyzing the second derivative.
Step 4: Find the y-intercept and any x-intercepts.
Step 5: Sketch the graph, including the information obtained from the previous steps.
Let's start with Step 1:
Step 1: Finding the first derivative:
The given function is f(x) = 2√(x) - x.
To find the first derivative, we differentiate with respect to x using the chain rule for the square root.
f'(x) = (2 * 1/2 * x^(-1/2)) - 1
= x^(-1/2) - 1
Now, let's move on to Step 2:
Step 2: Finding the critical points:
Critical points occur when the derivative equals 0.
Setting f'(x) = 0:
x^(-1/2) - 1 = 0
Solving for x:
x^(-1/2) = 1
Taking the square of both sides:
x^(-1) = 1^2
x^(-1) = 1
1/x = 1
x = 1
So, we have one critical point at x = 1.
Next, we move on to Step 3:
Step 3: Determining the nature of the critical point:
To determine the nature of the critical point, we will analyze the second derivative.
Taking the second derivative:
f''(x) = (1/2)(-1)(x^(-3/2))
= -1/(2x^(3/2))
Since the second derivative is negative, f''(1) = -1/(2√(1)), which is negative.
This means that the graph will be concave down at x = 1.
Moving on to Step 4:
Step 4: Finding the y-intercept and any x-intercepts:
To find the y-intercept, we substitute x = 0 into the original function:
f(0) = 2√(0) - 0
= 0 - 0
= 0
So, the y-intercept is (0, 0).
To find the x-intercept, we set the original function equal to 0 and solve for x:
2√(x) - x = 0
2√(x) = x
Squaring both sides:
4x = x^2
Rearranging the equation to find the x-intercepts:
x^2 - 4x = 0
x(x - 4) = 0
So, the x-intercepts are x = 0 and x = 4.
Finally, we move on to Step 5:
Step 5: Sketching the graph:
Using the information gathered from the previous steps, we can sketch the graph of the function.
- There is a y-intercept at (0, 0).
- There are x-intercepts at (0, 0) and (4, 0).
- There is a critical point at (1, f(1)), where f(1) can be calculated by substituting x = 1 into the original function.
Now, we can plot these points on the coordinate plane and sketch the graph based on the properties we determined:
Note: Since we have gathered enough information to draw the graph, it is not possible to show the graph through text. I recommend using a graphing calculator or plotting software to visualize the graph of the function.