STEP I 1990 question 9 solution

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Therefore the equation of the tangent at the point is

The co-ordinates of C are at the intersection of and

Using the expression for the area of a trapezium:

Area of ACC'A' =

Area of CBB'C' =

Sum of Areas =

As the gradient is constantly increasing (becoming 'less' negative), the area under the curve between 1 and b is greater than the sum of the areas of ACC'A' and CBB'C:

To prove the left-hand inequality, it is sufficient to show that

Which is clearly true.

The inequality follows immediately from the fact that by the Maclaurin expansion of .

Alternatively, note that the area of the trapezium ABB'A' is greater than the area under the curve between 1 and b.

Therefore

Solution by Dystopia.