Maximum and minimum points on a domain Problem set B

For \(y=x^{3} - 27 x + 4\), \(-4\le x\le 7\), use your calculator to find (i) any local maximum and local minimum points, (ii) the greatest value of y.

1. (a) Local maximum and minimum points.
2. (b) The greatest value of y.

For \(y=3 x^{4} - 12 x^{2} + 5\), \(-1\le x\le 2.5\), use your calculator to find (i) any local maximum and local minimum points, (ii) the greatest value of y.

1. (a) Local maximum and minimum points.
2. (b) The greatest value of y.

The curve \(y=x^{4} + b x^{2} + c\) has a horizontal tangent at the point (3, 4). Find the values of b and c.

Given that \(f(x)=x^{4} - x^{3} - 5 x + 1\), solve the equation

1. (a) \(f(x)=0\);
2. (b) \(f'(x)=0\).

Find the coordinates of the points on the graph of \(y=8x^{2}-x^{4}\) where the tangent is horizontal.

Find the coordinates of the local maximum point on the graph of \(y=4 x - \frac{5}{x^{2}}\).

A function is defined by \(f(x)=2 x^{3} - 5 x^{2} - 4 x + 3\).

1. (a) Find the coordinates of the local minimum point on the graph of \(y=f(x)\).
2. (b) Find the smallest value of the function on the domain \(-2\le x\le 4\).

Find the range of each function.

1. (a) \(f(x)=5 x + \frac{20}{x}\), for \(1\le x\le 6\);
2. (b) \(f(x)=3 x^{4} - 16 x^{3} + 18 x^{2}\), for all real x.

Let \(g(x)=x^{3} - 12 x^{2} + 36 x + 7\).

1. (a) Use your calculator to solve \(g'(x)=0\).
2. (b) Write down the interval on which g is decreasing.
3. (c) Find the local maximum point and the local minimum point of the graph of g.

The curve \(y=x^{3} + 6 x^{2} + 12 x - 2\) has a horizontal tangent at the point \((-2, -10)\).

1. (a) Show that the gradient of the curve is 0 at \(x=-2\).
2. (b) Find the gradient at \(x=-3\) and at \(x=-1\).
3. (c) State whether \((-2, -10)\) is a local maximum, a local minimum or neither. Give a reason.

The electricity demand, \(D\) megawatts, of a town is modelled by \(D=0.5t^3-14t^2+110t+20\), where t is the time in hours after 6 am, \(0\le t\le 18\). Use your calculator to find

1. (a) the greatest demand, and when it happens;
2. (b) the least demand, and when it happens.

The depth of a river bed, \(h\) metres below a bridge, at distance t hundred metres from the bank is modelled by \(h=at^3+bt^2-9t+31\). The depth has a local minimum at the point (3, 4). Find the values of a and b.

The monthly profit of a shop, P thousand euros, t months after it opens is modelled by \(P=0.1t^3-1.2t^2+3.6t+5\), for \(0\le t\le 8\).

1. (a) Find the values of t for which \(\frac{dP}{dt}=0\).
2. (b) Find the months during which the profit is falling.
3. (c) The profit has a local maximum and a local minimum in this model. Find the time and the profit of each.

The graph of \(y=3 x^{4} - 20 x^{3} + 12 x^{2} + 96 x\) has three points where the tangent is horizontal.

1. (a) Find the coordinates of the three points.
2. (b) Write down the range of the function \(f(x)=3 x^{4} - 20 x^{3} + 12 x^{2} + 96 x\) for all real x.

The curve \(y=ax^2+\frac{b}{x}\) has a horizontal tangent at the point (1, 12). Find the values of a and b.