Mixed practice: stationary points, concavity, parameters and models Problem set A

Sketch a graph of \(y=f(x)\) where \(f'(x)<0\) for all x, \(f''(x)<0\) for \(x<1\) and \(f''(x)>0\) for \(x>1\). Mark the point where the bend changes.

The curve \(y=x^3+px^2-2x+9\) is concave up for \(x>-1\) only.

1. (a) Find the value of p.
2. (b) What can you say about the curve at \(x=-1\)?

The function \(f(x)=x^n\) has \(f''(1)=20\). Find the possible values of n.

A curve has a minimum point. Find the values of b and c in each case.

1. (a) \(y=x^2+bx+c\) has a minimum at \((-2,5)\).
2. (b) \(y=x^4+bx+c\) has a minimum at \((2,-10)\).

Find the stationary points of each curve and classify them.

1. (a) \(y=2x-5\sqrt x\), for \(x>0\). Give exact values.
2. (b) \(y=\dfrac{x}{x^2+4}\). Give exact values.
3. (c) \(y=x+2\cos x\), for \(0<x<2\pi\). Give exact values.

Show each of these results.

1. (a) The graph of \(y=3x^2+bx+c\) is concave up for every value of x.
2. (b) If \(y=3\sin 2x+5\cos 2x\), then \(\dfrac{d^2y}{dx^2}=-4y\).
3. (c) The graph of \(y=\ln(a+2x)\) is always concave down.

The curve is \(y=x^3-9x^2+15x+4\).

1. (a) Find the interval in which the gradient is decreasing.
2. (b) Hence find the coordinates of the point of inflection.

The curve is \(y=x^4\ln x\), for \(x>0\).

1. (a) Find the x-coordinate of the point where \(y''=2\). Use the calculator to solve the equation.
2. (b) The curve has a point of inflection. Find its x-coordinate.

The function f is defined by \(f(x)=\dfrac{\ln x}{x^3}\), for \(x>0\).

1. (a) Find \(f'(x)\).
2. (b) Find the exact coordinates of the stationary point.
3. (c) Show that \(f''(x)=\dfrac{12\ln x-7}{x^5}\).
4. (d) Hence classify the stationary point.

A bakery makes x hundred loaves a day, where \(x>0\). The daily profit, in hundreds of dollars, is modelled by \(P=-x^3+12x^2-21x-30\).

1. (a) Find the value of x that gives the greatest profit. Show that it is a maximum.
2. (b) Find the greatest profit.

Water runs into a garden tank. The volume of water, V litres, after t hours is modelled by \(V=t^3-6t^2+17t\), for \(t\ge0\). Find the time at which the tank is filling most slowly, and state the minimum rate of filling.

The number of seabirds on an island, P thousand, is modelled by \(P=\dfrac{12t^2}{9+t^2}\), where t is the time in years since monitoring began.

1. (a) Find \(\dfrac{dP}{dt}\) and use it to find the time at which the number is increasing most quickly. Give your answer to 3 significant figures.
2. (b) Find the number at this time and the greatest rate of increase.

Find the stationary points of the curve \(y=px+\dfrac{4}{x}\), where \(p>0\), and classify them. Give your answers in terms of p.

A cuboid has a square base of side x cm and height \((a-2x)\) cm, where a is a positive constant and \(0<x<\dfrac{a}{2}\).

1. (a) Find the value of x, in terms of a, that gives the greatest volume. Justify that it is a maximum.
2. (b) Find the greatest volume in terms of a.

Answer both parts.

1. (a) Find the range of \(f(x)=e^{2x}-ax\), where \(a>0\), in terms of a.
2. (b) A model for the potential energy U joules of a particle at position x metres is \(U=e^{2x}+e^{-3x}\). Find the exact minimum value of U.

Answer both parts.

1. (a) The curve \(y=x^3-3x^2+bx+2\) has a positive gradient at every point (it is strictly increasing and never flat). Find the condition on b.
2. (b) The curve \(y=ax^3-bx^2\), where a and b are positive, is increasing for \(x<0\) and for \(x>6\). Evaluate \(\dfrac{b}{a}\).

A curve has \(y''=6x+4\). Its gradient at \(x=0\) is −2 and it passes through the point \((1,5)\). Find y in terms of x.

The graph shows \(y=f'(x)\). It crosses the x-axis at \(x=-2\) and touches the x-axis at \(x=1\).

1. (a) Write down the x-coordinate of each local minimum of \(f\), and of each local maximum of \(f\).
2. (b) Write down the x-coordinate of each point of inflection of \(f\).
3. (c) Which of these points of inflection are stationary points of inflection? Explain.