Find a point P on curve y = x^3 such that the slope of the line passing through point P & (1,1) is 3/4.

To find a point P on the curve y = x^3 such that the slope of the line passing through point P and (1, 1) is 3/4, we need to use the following steps:

Step 1: Express the equation of the curve y = x^3 in terms of x and y.

The equation y = x^3 implies that y is equal to the cube of x. So, we can rewrite the equation as y^3 = x.

Step 2: Use the slope formula to determine the slope of the line passing through the points P and (1, 1).

The slope formula is given by:
slope = (y2 - y1) / (x2 - x1)

Let the coordinates of point P be (x, y). Using the given point (1, 1), we can substitute the values into the slope formula:
3/4 = (y - 1) / (x - 1)

Step 3: Substitute the expression for y from the equation of the curve into the slope formula.

Since we already have the equation y^3 = x, we can substitute y from this equation back into the slope formula:
3/4 = (x^3 - 1) / (x - 1)

Step 4: Solve the resulting equation to find the value(s) of x.

To solve the equation 3/4 = (x^3 - 1) / (x - 1) for x, we need to cross-multiply:
3(x - 1) = 4(x^3 - 1)

Expanding the equation gives us:
3x - 3 = 4x^3 - 4

Rearranging the terms:
4x^3 - 3x - 4x + 1 = 0

Combining like terms:
4x^3 - 7x + 1 = 0

This equation is a cubic equation and can be challenging to solve algebraically. We can use numerical methods or graphing calculators to find the approximate values of x.

Step 5: Substitute the found value(s) of x back into the equation y = x^3 to find the corresponding value(s) of y.

Once we have the value(s) of x, we can substitute them back into the equation y = x^3 to find the corresponding value(s) of y.

This process will give us the point(s) P (x, y) on the curve y = x^3 that satisfy the given conditions.