Increasing/decreasing intervals

3. $$y = x ^ { 2 } - k \sqrt { } x , \text { where } k \text { is a constant. }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Given that \(y\) is decreasing at \(x = 4\), find the set of possible values of \(k\).

7 In this question you must show detailed reasoning.
A curve has equation \(y = 4 x ^ { 3 } - 6 x ^ { 2 } - 9 x + 4\).

1. Sketch the gradient function for this curve, clearly indicating the points where the gradient is zero.
2. Find the set of values of \(x\) for which the gradient function is decreasing. Give your answer using set notation.

8 The gradient, \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), at the point \(( x , y )\) on a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 54 + 27 x - 6 x ^ { 2 }$$

1. 1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
   2. The curve passes through the point \(P \left( - 1 \frac { 1 } { 2 } , 4 \right)\). Verify that the curve has a minimum point at \(P\).
2. 1. Show that at the points on the curve where \(y\) is decreasing $$2 x ^ { 2 } - 9 x - 18 > 0$$
   2. Solve the inequality \(2 x ^ { 2 } - 9 x - 18 > 0\).

8 The function f is given by \(\mathrm { f } ( x ) = \frac { x ^ { 2 } } { 3 x ^ { 2 } - 1 }\), for \(x > 1\). Show that f is a decreasing function.

5

1. Differentiate \(\frac { x } { \sqrt { 1 + x ^ { 2 } } }\) with respect to \(x\).
2. Hence show that \(\frac { x } { \sqrt { 1 + x ^ { 2 } } }\) is increasing for all \(x\).

\includegraphics{figure_11} A function is defined by f\((x) = \frac{4}{x^3} - \frac{3}{x} + 2\) for \(x \neq 0\). The graph of \(y = \text{f}(x)\) is shown in the diagram.

1. Find the set of values of \(x\) for which f\((x)\) is decreasing. [5]
2. A triangle is bounded by the \(y\)-axis, the normal to the curve at the point where \(x = 1\) and the tangent to the curve at the point where \(x = -1\). Find the area of the triangle. Give your answer correct to 3 significant figures. [8]

A function f is such that \(f'(x) = 6(2x-3)^2 - 6x\) for \(x \in \mathbb{R}\).

1. Determine the set of values of \(x\) for which f\((x)\) is decreasing. [4]
2. Given that f\((1) = -1\), find f\((x)\). [4]

Differentiate \(2x^3 + 9x^2 - 24x\). Hence find the set of values of \(x\) for which the function \(f(x) = 2x^3 + 9x^2 - 24x\) is increasing. [4]

Use calculus to find the set of values of \(x\) for which \(\text{f}(x) = 12x - x^3\) is an increasing function. [3]

Differentiate \(2x^3 + 9x^2 - 24x\). Hence find the set of values of \(x\) for which the function \(f(x) = 2x^3 + 9x^2 - 24x\) is increasing. [4]

Find the set of values of \(x\) for which \(x^2 - 7x\) is a decreasing function. [3]

The gradient of a curve is given by \(\frac{dy}{dx} = x^2 - 6x\). Find the set of values of \(x\) for which \(y\) is an increasing function of \(x\). [3]

The function f, defined for \(x \in \mathbb{R}, x > 0\), is such that $$f'(x) = x^2 - 2 + \frac{1}{x^2}$$

1. Find the value of f''(x) at \(x = 4\). [3]
2. Given that f(3) = 0, find f(x). [4]
3. Prove that f is an increasing function. [3]

Chris claims that, "for any given value of \(x\), the gradient of the curve \(y = 2x^3 + 6x^2 - 12x + 3\) is always greater than the gradient of the curve \(y = 1 + 60x - 6x^2\)". Show that Chris is wrong by finding all the values of \(x\) for which his claim is not true. [7 marks]

Prove that the function \(f(x) = x^3 - 3x^2 + 15x - 1\) is an increasing function. [6 marks]

A curve has equation \(y = \frac{2x + 3}{4x^2 + 7}\)

1. 1. Find \(\frac{dy}{dx}\) [2 marks]
   2. Hence show that \(y\) is increasing when \(4x^2 + 12x - 7 < 0\) [4 marks]
2. Find the values of \(x\) for which \(y\) is increasing. [2 marks]

Prove that \(f(x) = x^3 - 6x^2 + 13x - 7\) is an increasing function. [4]

A curve has equation $$y = 2x^3 - 2x^2 - 2x + 8$$

1. Find \(\frac{dy}{dx}\) [2]
2. Hence find the range of values of \(x\) for which \(y\) is increasing. Write your answer in set notation. [4]

The function \(f\) is defined by $$f(x) = \frac{(x + 5)(x + 1)}{(x + 4)} - \ln(x + 4) \quad x \in \mathbb{R} \quad x > k$$

1. State the smallest possible value of \(k\). [1]
2. Show that $$f'(x) = \frac{ax^2 + bx + c}{(x + 4)^2}$$ where \(a\), \(b\) and \(c\) are integers to be found. [4]
3. Hence show that \(f\) is an increasing function. [2]

In this question you must show detailed reasoning. Find the values of \(x\) for which the gradient of the curve \(y = \frac{2}{3}x^3 + \frac{5}{2}x^2 - 3x + 7\) is positive. Give your answer in set notation. [5]