CIE/GCE/AS/Math/P1/19/Nov/13/Q#3
Question
The equation of a curve is 𝑦 = 𝑥 3 + 𝑥 2 − 8𝑥 + 7. The curve has no stationary points in
the interval 𝑎 < 𝑥 < 𝑏. Find the least possible value of 𝑎 and the greatest possible
value of 𝑏.
Solution
We are given;
𝑦 = 𝑥 3 + 𝑥 2 − 8𝑥 + 7
We are given that curve has no stationary point.
A stationary point 𝑆 (𝑥, 𝑦) on the curve 𝑦 is the point where gradient of the curve is
equal to zero;
𝑑𝑦
𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡 𝑜𝑓 𝑡ℎ𝑒 𝐶𝑢𝑟𝑣𝑒𝐴𝑡 𝑆𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑟𝑦 𝑃𝑜𝑖𝑛𝑡 = | = 0
𝑑𝑥 𝑠
𝑑𝑦
Therefore, for the given curve the = 0 should have no solution.
𝑑𝑥
𝑑𝑦
First, we find .
𝑑𝑥
Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient
of curve 𝑦 with respect to 𝑥 is:
𝑑𝑦
𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡 =
𝑑𝑥
𝑑𝑦 𝑑 3
= (𝑥 + 𝑥 2 − 8𝑥 + 7)
𝑑𝑥 𝑑𝑥
Rule for differentiation of 𝑦 = 𝑔(𝑥 ) + ℎ(𝑥) is:
𝑑 𝑑 𝑑
[𝑔(𝑥 ) + ℎ(𝑥)] = [𝑔(𝑥)] + [ℎ(𝑥)]
𝑑𝑥 𝑑𝑥 𝑑𝑥
Question
The equation of a curve is 𝑦 = 𝑥 3 + 𝑥 2 − 8𝑥 + 7. The curve has no stationary points in
the interval 𝑎 < 𝑥 < 𝑏. Find the least possible value of 𝑎 and the greatest possible
value of 𝑏.
Solution
We are given;
𝑦 = 𝑥 3 + 𝑥 2 − 8𝑥 + 7
We are given that curve has no stationary point.
A stationary point 𝑆 (𝑥, 𝑦) on the curve 𝑦 is the point where gradient of the curve is
equal to zero;
𝑑𝑦
𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡 𝑜𝑓 𝑡ℎ𝑒 𝐶𝑢𝑟𝑣𝑒𝐴𝑡 𝑆𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑟𝑦 𝑃𝑜𝑖𝑛𝑡 = | = 0
𝑑𝑥 𝑠
𝑑𝑦
Therefore, for the given curve the = 0 should have no solution.
𝑑𝑥
𝑑𝑦
First, we find .
𝑑𝑥
Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient
of curve 𝑦 with respect to 𝑥 is:
𝑑𝑦
𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡 =
𝑑𝑥
𝑑𝑦 𝑑 3
= (𝑥 + 𝑥 2 − 8𝑥 + 7)
𝑑𝑥 𝑑𝑥
Rule for differentiation of 𝑦 = 𝑔(𝑥 ) + ℎ(𝑥) is:
𝑑 𝑑 𝑑
[𝑔(𝑥 ) + ℎ(𝑥)] = [𝑔(𝑥)] + [ℎ(𝑥)]
𝑑𝑥 𝑑𝑥 𝑑𝑥
