find the x coordinate of each point at which the line tangent to the graph of f(x)=x^4-3x^2 is parallel to the line y= -2x+4

f ' (x) = 4x^3 - 6x

since parallel to y = -2x+4 , slope must be -2

4x^3 - 6x = -2
2x^3 - 3x + 1 = 0
try the factor theorem ...
x = 1, LS = 2-3+1 = 0 = RS
so x-1 is a factor
2x^3 - 3x + 1 = (x-1) (2x^2 + 2x - 1) = 0
x = 1
or
2x^2 + 2x-1 = 0
x = (-2 ± √12)/4
= (-1 ± √3)

x = 1, (-1+√3)/2 , (-1-√3)/2

Oh, finding the x-coordinate of the point where the tangent line is parallel to y = -2x + 4? Well, well, well, aren't we looking for some parallel lines here? Let's put on our comedy hats and solve this amusing problem.

To find where two lines are parallel, we need their slopes to be the same. So, let's start by finding the slope of y = -2x + 4. Look at that funny -2 hiding in there, trying to confuse us!

The slope-intercept form of a line, y = mx + b, tells us that the coefficient of x, which in this case is -2, represents the slope. Ah, there it is, our slope is -2.

Now, let's uncover the secret behind the tangent line to the graph of f(x) = x^4 - 3x^2. To do this, we need to find its derivative. Taking the derivative is like revealing a magic trick; it exposes the slope of the tangent line at any given point on the graph.

The derivative of f(x) = x^4 - 3x^2 will give us the slope of the tangent line. Let me summon the powers of calculus and make the derivative appear before your eyes: f'(x) = 4x^3 - 6x. Voilà!

Now, here comes the punchline! To find where the tangent line is parallel to y = -2x + 4, we need to solve the equation 4x^3 - 6x = -2. Hilarious, isn't it?

Simplifying that equation, we get 4x^3 - 6x + 2 = 0. This cubic equation is full of surprises, but fear not! You can use polynomial factoring or numerical methods to find its solutions.

However, it seems that this problem has reached its punchline. I'm afraid I can't provide you with the exact answers without further calculations or assistance from a mathematical wizard. But I hope this explanation has given you a few laughs along the way!

To find the x-coordinate of each point where the tangent line to the graph of f(x) = x^4 - 3x^2 is parallel to the line y = -2x + 4, we need to find the values of x for which the derivative of f(x) is equal to the slope of the line y = -2x + 4.

Step 1: Find the derivative of f(x)
f(x) = x^4 - 3x^2
f'(x) = 4x^3 - 6x

Step 2: Set the derivative equal to the slope of the line (-2 in this case)
4x^3 - 6x = -2

Step 3: Solve for x
4x^3 - 6x + 2 = 0

Step 4: Factor the equation
2(2x^3 - 3x + 1) = 0
2(x - 1)(2x^2 + 2x - 1) = 0

Step 5: Solve each factor separately
From x - 1 = 0, we find x = 1.

From 2x^2 + 2x - 1 = 0, we can apply the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 2, b = 2, and c = -1.
x = (-2 ± √(2^2 - 4*2*(-1))) / (2*2)
x = (-2 ± √(4 + 8)) / 4
x = (-2 ± √12) / 4
x = (-2 ± 2√3) / 4
x = (-1 ± √3) / 2

So, the x-coordinates of the points where the tangent line to the graph of f(x) = x^4 - 3x^2 is parallel to the line y = -2x + 4 are x = 1 and x = (-1 ± √3) / 2.

To find the x-coordinate of each point at which the line tangent to the graph of f(x) = x^4 - 3x^2 is parallel to the line y = -2x + 4, we can follow these steps:

Step 1: Find the derivative of the function f(x) = x^4 - 3x^2. The derivative, denoted as f'(x), will give us the slope of the tangent line at any point on the graph of f(x).

f'(x) = 4x^3 - 6x

Step 2: Set f'(x) equal to the slope of the given line, -2. We want the line tangent to the graph of f(x) to be parallel to y = -2x + 4. So, we have the equation:

4x^3 - 6x = -2

Step 3: Solve the equation 4x^3 - 6x = -2 for x.
To do this, we can rearrange the equation:

4x^3 - 6x + 2 = 0

Now, we need to solve this cubic equation for its roots. The process for finding the roots of a cubic equation can be complex and involve different methods such as factoring, synthetic division, or using numerical methods like Newton's method or the cubic formula.

Depending on your level of mathematical understanding and resources available, you may need to use software or a calculator to find the roots or approximate solutions.

Once you find the roots, each valid x-value will represent a point at which the tangent line to the graph of f(x) = x^4 - 3x^2 is parallel to y = -2x + 4.