At what point of the graph of x^2 / 2 is the tangent line parallel to the line 2x-4y = 3
TIA
To find the point on the graph of x^2 / 2 where the tangent line is parallel to the line 2x - 4y = 3, we can first determine the slope of the line 2x - 4y = 3, and then find the derivative of the function x^2 / 2.
Step 1: Finding the slope of the line
The given line can be rewritten in slope-intercept form (y = mx + c), where m is the slope:
2x - 4y = 3
-4y = -2x + 3
y = (1/2)x - 3/4
Comparing this equation to y = mx + c, we see that the slope of the line is 1/2.
Step 2: Finding the derivative of the function
The function given is f(x) = x^2 / 2. We need to find its derivative f'(x) to determine the slope of the tangent line at any given point.
To find the derivative, we can apply the power rule:
f(x) = x^2 / 2
f'(x) = d/dx (x^2 / 2)
= 1/2 * d/dx (x^2)
= 1/2 * (2x)
= x
So, the derivative of f(x) = x^2 / 2 is f'(x) = x.
Step 3: Finding the point of tangency
Since we want the tangent line to be parallel to the line 2x - 4y = 3, the slope of the tangent line should also be 1/2. Therefore, we need to find the x-coordinate(s) at which f'(x) = x equals 1/2.
Setting x = 1/2, we have:
1/2 = x^2 / 2
1 = x^2
x = ±1
So, the x-coordinates at which the tangent line is parallel to the line 2x - 4y = 3 are x = 1 and x = -1. To find the corresponding y-coordinates, we substitute these x-values back into the original equation f(x) = x^2 / 2:
When x = 1:
f(1) = (1)^2 / 2
= 1/2
When x = -1:
f(-1) = (-1)^2 / 2
= 1/2
Therefore, the points of the graph x^2 / 2 at which the tangent line is parallel to the line 2x - 4y = 3 are (1, 1/2) and (-1, 1/2).