Functions and Graphs - Estimating Gradients from Tangents

The variables \(x\) and \(y\) are connected by the equation \(y = 3x^2 + \frac{5}{x}\).

Some corresponding values of \(x\) and \(y\) are given in the following table.

1. Find the value of \(p\).
2. Using a scale of \(1\) cm to \(0.1\) unit, draw a horizontal \(x\)-axis for \(0 \le x \le 1.6\). Using a scale of \(1\) cm to \(1\) unit, draw a vertical \(y\)-axis for \(0 \le y \le 14\).
   On your axes, plot the points given in the table and join them with a smooth curve.
3. Use your graph to find the least value of \(y\).
4. Use your graph to find two solutions of \(3x^2 + \frac{5}{x} = 9\) in the range \(0 \le x \le 1.6\).
5. By drawing a tangent, find the gradient of the curve at the point \((0.8, 8.17)\).
6. 1. On the same axes, draw the graph of \(y = 2x + 7\) for \(0 \le x \le 1.6\).
   2. Write down \(x\)-coordinate of the point where the two graphs intersect.
   3. This value of \(x\) is a solution of the equation \(3x^3 + Ax^2 + Bx + C = 0\).
      Find the value of \(A\), \(B\) and \(C\).

Use \(x>0\): the function is undefined at \(x=0\). In (f)(ii), there are two intersections in the stated positive interval, so read both coordinates rather than one.