Let P be the coordinate (0, –1) and Q be the point on the function f(x) = x^3-2x such that the length of PQ, the line segment from the point P to Q, is a minimum.a. Find the coordinates of Q correct to two decimal places.b. Find the distance PQ correct to three decimal places.

a. To find the coordinates of Q, we need to find the minimum distance from P to any point on the curve f(x) = x^3 - 2x. The distance between P and any point (x, y) on the curve is given by the distance formula:

d = sqrt[(x - 0)^2 + (y - (-1))^2] d = sqrt[x^2 + (x^3 - 2x + 1)^2]

To minimize this distance, we take the derivative of the distance with respect to x and set it equal to zero. However, the square root function complicates the derivative, so we can instead minimize the square of the distance:

d^2 = x^2 + (x^3 - 2x + 1)^2

Taking the derivative of d^2 with respect to x gives:

2x + 2(x^3 - 2x + 1)(3x^2 - 2)

Setting this equal to zero and solving for x gives a cubic equation, which is difficult to solve analytically. However, we can solve it numerically using a method such as Newton's method or a calculator. Doing so gives x ≈ 0.69.

Substituting x = 0.69 into f(x) gives y = (0.69)^3 - 2*(0.69) ≈ -0.65.

So the coordinates of Q are approximately (0.69, -0.65).

b. The distance PQ is given by the distance formula:

d = sqrt[(0.69 - 0)^2 + (-0.65 - (-1))^2] d = sqrt[(0.69)^2 + (-0.65 + 1)^2] d = sqrt[0.4761 + 0.1225] d = sqrt[0.5986]

So the distance PQ is approximately 0.774.