The function f is defined by \(f(x) = 2 - \frac{3}{4x-p}\) for \(x > \frac{p}{4}\), where \(p\) is a constant.

Find \(f'(x)\) and hence determine whether \(f\) is an increasing function, a decreasing function or neither.

A function \(f\) is defined by \(f : x \mapsto x^3 - x^2 - 8x + 5\) for \(x < a\). It is given that \(f\) is an increasing function. Find the largest possible value of the constant \(a\).

The function \(f\) is such that \(f(x) = x^3 - 3x^2 - 9x + 2\) for \(x > n\), where \(n\) is an integer. It is given that \(f\) is an increasing function. Find the least possible value of \(n\).

(i) Express \(3x^2 - 6x + 2\) in the form \(a(x+b)^2 + c\), where \(a, b\) and \(c\) are constants.

(ii) The function \(f\), where \(f(x) = x^3 - 3x^2 + 7x - 8\), is defined for \(x \in \mathbb{R}\). Find \(f'(x)\) and state, with a reason, whether \(f\) is an increasing function, a decreasing function or neither.

The function f is defined by \(f(x) = \frac{1}{x+1} + \frac{1}{(x+1)^2}\) for \(x > -1\).

1. Find \(f'(x)\).
2. State, with a reason, whether f is an increasing function, a decreasing function or neither.

The function g is defined by \(g(x) = \frac{1}{x+1} + \frac{1}{(x+1)^2}\) for \(x < -1\).

1. Find the coordinates of the stationary point on the curve \(y = g(x)\).

(i) Express \(9x^2 - 12x + 5\) in the form \((ax + b)^2 + c\).

(ii) Determine whether \(3x^3 - 6x^2 + 5x - 12\) is an increasing function, a decreasing function or neither.

The equation of a curve is \(y = x^3 + ax^2 + bx\), where \(a\) and \(b\) are constants.

(i) In the case where the curve has no stationary point, show that \(a^2 < 3b\).

(ii) In the case where \(a = -6\) and \(b = 9\), find the set of values of \(x\) for which \(y\) is a decreasing function of \(x\).

A function \(f\) is defined by \(f(x) = \frac{5}{1 - 3x}\), for \(x \geq 1\).

(i) Find an expression for \(f'(x)\).

(ii) Determine, with a reason, whether \(f\) is an increasing function, a decreasing function or neither.

It is given that \(f(x) = (2x - 5)^3 + x\), for \(x \in \mathbb{R}\). Show that \(f\) is an increasing function.

It is given that \(f(x) = \frac{1}{x^3} - x^3\), for \(x > 0\). Show that \(f\) is a decreasing function.

A curve has equation \(y = f(x)\). It is given that \(f'(x) = 3x^2 + 2x - 5\).

Find the set of values of \(x\) for which \(f\) is an increasing function.

It is given that a curve has equation \(y = k(3x-k)^{-1} + 3x\), where \(k\) is a constant.

(a) Find, in terms of \(k\), the values of \(x\) at which there is a stationary point.

The function \(f\) has a stationary value at \(x = a\) and is defined by \(f(x) = 4(3x-4)^{-1} + 3x\) for \(x \geq \frac{3}{2}\).

(b) Find the value of \(a\) and determine the nature of the stationary value.

(c) The function \(g\) is defined by \(g(x) = -(3x+1)^{-1} + 3x\) for \(x \geq 0\).

Determine, making your reasoning clear, whether \(g\) is an increasing function, a decreasing function or neither.

The equation of a curve is \(y = \frac{1}{6}(2x - 3)^3 - 4x\).

(i) Find \(\frac{dy}{dx}\).

(ii) Find the equation of the tangent to the curve at the point where the curve intersects the y-axis.

(iii) Find the set of values of \(x\) for which \(\frac{1}{6}(2x - 3)^3 - 4x\) is an increasing function of \(x\).

The function \(f\) is such that \(f(x) = \frac{3}{2x+5}\) for \(x \in \mathbb{R}, x \neq -2.5\).

Obtain an expression for \(f'(x)\) and explain why \(f\) is a decreasing function.

The function f is such that \(f(x) = (3x + 2)^3 - 5\) for \(x \geq 0\).

Obtain an expression for \(f'(x)\) and hence explain why f is an increasing function.

The diagram shows the graph of \(y = f(x)\), where \(f : x \mapsto \frac{6}{2x+3}\) for \(x \geq 0\).

Find an expression, in terms of \(x\), for \(f'(x)\) and explain how your answer shows that \(f\) is a decreasing function.

The diagram shows the curve \(y = x^3 - 3x^2 - 9x + k\), where \(k\) is a constant. The curve has a minimum point on the \(x\)-axis.

(i) Find the value of \(k\).

(ii) Find the coordinates of the maximum point of the curve.

(iii) State the set of values of \(x\) for which \(x^3 - 3x^2 - 9x + k\) is a decreasing function of \(x\).

A function f is defined by f : x ↦ (2x − 3)³ − 8, for 2 ≤ x ≤ 4.

Find an expression, in terms of x, for f'(x) and show that f is an increasing function.

The function \(f\) is defined by \(f(x) = x^5 - 10x^3 + 50x\) for \(x \in \mathbb{R}\).

Determine whether \(f\) is an increasing function, a decreasing function or neither.

The function \(f\) is defined by \(f(x) = \frac{1}{3}(2x - 1)^{\frac{3}{2}} - 2x\) for \(\frac{1}{2} < x < a\). It is given that \(f\) is a decreasing function.

Find the maximum possible value of the constant \(a\).

The function \(f\) is defined by \(f(x) = \frac{1}{3x+2} + x^2\) for \(x < -1\).

Determine whether \(f\) is an increasing function, a decreasing function or neither.

An increasing function, \(f\), is defined for \(x > n\), where \(n\) is an integer. It is given that \(f'(x) = x^2 - 6x + 8\). Find the least possible value of \(n\).

The function \(f\) is defined by \(f'(x) = x^3 + 2x^2 - 4x + 7\) for \(x \geq -2\). Determine, showing all necessary working, whether \(f\) is an increasing function, a decreasing function or neither.

(i) The tangent to the curve \(y = x^3 - 9x^2 + 24x - 12\) at a point \(A\) is parallel to the line \(y = 2 - 3x\). Find the equation of the tangent at \(A\).

(ii) The function \(f\) is defined by \(f(x) = x^3 - 9x^2 + 24x - 12\) for \(x > k\), where \(k\) is a constant. Find the smallest value of \(k\) for \(f\) to be an increasing function.

The function \(f\) is such that \(f(x) = (2x - 1)^{\frac{3}{2}} - 6x\) for \(\frac{1}{2} < x < k\), where \(k\) is a constant. Find the largest value of \(k\) for which \(f\) is a decreasing function.