To find the equation of the curve, we need to integrate the gradient function \(\frac{dy}{dx} = x - 3x^{-\frac{1}{2}}\).

Integrate term by term:

\(\int x \, dx = \frac{x^2}{2}\)

\(\int -3x^{-\frac{1}{2}} \, dx = -3 \cdot \frac{x^{\frac{1}{2}}}{\frac{1}{2}} = -6x^{\frac{1}{2}}\)

Thus, the integral is:

\(y = \frac{x^2}{2} - 6x^{\frac{1}{2}} + c\)

We use the point \((4, 1)\) to find \(c\):

\(1 = \frac{4^2}{2} - 6 \cdot 4^{\frac{1}{2}} + c\)

\(1 = 8 - 12 + c\)

\(1 = -4 + c\)

\(c = 5\)

Therefore, the equation of the curve is:

\(y = \frac{1}{2}x^2 - 6x^{\frac{1}{2}} + 5\)

(a) To find the equation of the tangent, we first find the gradient at \(x = 6\) using \(\frac{dy}{dx} = 12\left(\frac{1}{2}x - 1\right)^{-4}\).

Substitute \(x = 6\):

\(12\left(\frac{1}{2} \times 6 - 1\right)^{-4} = 12(2)^{-4} = \frac{3}{4}\)

The gradient of the tangent is \(\frac{3}{4}\). Using the point-slope form \(y - y_1 = m(x - x_1)\) with \(P(6, 4)\):

\(y - 4 = \frac{3}{4}(x - 6)\)

Simplifying gives:

\(y = \frac{3}{4}x - \frac{1}{2}\)

(b) To find the equation of the curve, integrate \(\frac{dy}{dx} = 12\left(\frac{1}{2}x - 1\right)^{-4}\).

\(y = \int 12\left(\frac{1}{2}x - 1\right)^{-4} \, dx\)

Let \(u = \frac{1}{2}x - 1\), then \(du = \frac{1}{2}dx\) or \(dx = 2du\).

\(y = \int 12u^{-4} \, 2du = 24\int u^{-4} \, du\)

\(y = 24\left(\frac{u^{-3}}{-3}\right) + c = -8u^{-3} + c\)

Substitute back \(u = \frac{1}{2}x - 1\):

\(y = -8\left(\frac{1}{2}x - 1\right)^{-3} + c\)

Using point \(P(6, 4)\) to find \(c\):

\(4 = -8(2)^{-3} + c\)

\(4 = -1 + c\)

\(c = 5\)

Thus, the equation of the curve is:

\(y = -8\left(\frac{1}{2}x - 1\right)^{-3} + 5\)

To find the equation of the curve, we need to integrate \(\frac{dy}{dx} = 3(4x - 7)^{\frac{1}{2}} - 4x^{-\frac{1}{2}}\).

Integrating the first term:

\(\int 3(4x - 7)^{\frac{1}{2}} \, dx = \frac{3}{2 \times 4} (4x - 7)^{\frac{3}{2}}\)

Integrating the second term:

\(\int -4x^{-\frac{1}{2}} \, dx = -8x^{\frac{1}{2}}\)

Thus, the integrated function is:

\(y = \frac{3}{6}(4x - 7)^{\frac{3}{2}} - 8x^{\frac{1}{2}} + c\)

Using the point \((4, \frac{5}{2})\) to find \(c\):

\(\frac{5}{2} = \frac{1}{2}(9)^{\frac{3}{2}} - 8 \times 4^{\frac{1}{2}} + c\)

\(\frac{5}{2} = \frac{1}{2}(27) - 8 \times 2 + c\)

\(\frac{5}{2} = \frac{27}{2} - 16 + c\)

\(\frac{5}{2} = \frac{27}{2} - \frac{32}{2} + c\)

\(\frac{5}{2} = -\frac{5}{2} + c\)

\(c = 5\)

Therefore, the equation of the curve is:

\(y = \frac{3}{6}(4x - 7)^{\frac{3}{2}} - 8x^{\frac{1}{2}} + 5\)

(a) To determine the nature of the stationary point, evaluate \(\frac{d^2y}{dx^2}\) at \(x = -1\):

\(\frac{d^2y}{dx^2} = 6(-1)^2 - \frac{4}{(-1)^3} = 6 \times 1 + 4 = 10\).

Since \(\frac{d^2y}{dx^2} > 0\), the stationary point is a minimum.

(b) Integrate \(\frac{d^2y}{dx^2}\) to find \(\frac{dy}{dx}\):

\(\frac{dy}{dx} = \int (6x^2 - \frac{4}{x^3}) \, dx = 2x^3 + \frac{2}{x^2} + c\).

Substitute \(x = -1\) and \(\frac{dy}{dx} = 0\) to find \(c\):

\(0 = 2(-1)^3 + \frac{2}{(-1)^2} + c \Rightarrow c = 0\).

Integrate \(\frac{dy}{dx}\) to find \(y\):

\(y = \int (2x^3 + \frac{2}{x^2}) \, dx = \frac{1}{2}x^4 - \frac{2}{x} + k\).

Substitute \(x = -1\), \(y = \frac{9}{2}\) to find \(k\):

\(\frac{9}{2} = \frac{1}{2}(-1)^4 - \frac{2}{-1} + k \Rightarrow k = 2\).

Thus, the equation of the curve is \(y = \frac{1}{2}x^4 - \frac{2}{x} + 2\).

(c) Set \(\frac{dy}{dx} = 0\):

\(2x^3 + \frac{2}{x^2} = 0 \Rightarrow x^5 = -1\).

The only real solution is \(x = -1\), so there are no other stationary points.

(d) At \(x = 1\), \(\frac{dy}{dx} = 4\).

Using the chain rule, \(\frac{dx}{dt} = \frac{dx}{dy} \cdot \frac{dy}{dt} = \frac{1}{4} \times 5 = \frac{5}{4}\).

To find \(f(x)\), we need to integrate \(f'(x) = 2x^{-\frac{1}{3}} - x^{\frac{1}{3}}\).

Integrating term by term:

\(\int 2x^{-\frac{1}{3}} \, dx = 2 \cdot \frac{x^{\frac{2}{3}}}{\frac{2}{3}} = 3x^{\frac{2}{3}}\)

\(\int x^{\frac{1}{3}} \, dx = \frac{x^{\frac{4}{3}}}{\frac{4}{3}} = \frac{3}{4}x^{\frac{4}{3}}\)

Thus, \(f(x) = 3x^{\frac{2}{3}} - \frac{3}{4}x^{\frac{4}{3}} + c\).

Using the condition \(f(8) = 5\):

\(5 = 3(8)^{\frac{2}{3}} - \frac{3}{4}(8)^{\frac{4}{3}} + c\)

\(5 = 12 - 12 + c\)

\(c = 5\)

Therefore, \(f(x) = 3x^{\frac{2}{3}} - \frac{3}{4}x^{\frac{4}{3}} + 5\).