Extremum problems with rational and root functions · Grade 11 · 4 units
In Extremum problems with rational and root functions (Grade 11 · 4 units) you practise 5 skills: Building the function · Problems with a rational function · Problems with a root function · Problems with a box · Greatest and least values on a closed interval. An extremum problem starts with choosing a variable: write the quantity to be made greatest or least in terms of it, and state the domain the variable can move in. In these problems the function can be rational, such as a number plus a multiple of its reciprocal, or have a root, such as the distance between two points.
Then differentiate and set the derivative to 0. Where the derivative is 0 and changes sign there is a turning point: a minimum when it goes from negative to positive, and a maximum when it goes from positive to negative. When a root sits in the denominator of the derivative it is positive, so the numerator decides the sign. On a closed interval the greatest or least value can also be at an end of the interval, so compare the value at the turning point with the values at the ends. In problems with a box, write the height with the side of the base and the given volume, and the surface area is the sum of the areas of all its faces.
Tip: At the end of every problem, go back to the question: sometimes it asks about the variable, and sometimes about the value of the quantity.
What you learn
Building the function
For example: The perimeter of a rectangle is 14 , and one of its sides is x . Write its area as a function of x .
Answer and solution
The answer: −x² + 7x
Two neighbouring sides add up to half the perimeter, 7 , so the other side is 7 − x . The area: A(x) = x(7 − x) = −x² + 7x
Problems with a rational function
For example: The product of two positive numbers is 9 . What is the least possible sum of the two numbers?
Answer and solution
The answer: 6
One number is x , and the other is 9 / x . The sum: S(x) = x + 9 / x The domain: x > 0 S′(x) = 1 − 9 / x² = 0 so x² = 9 and x = 3 . For x < 3 the derivative is negative, and for x > 3 it is positive. This is a minimum point. The other number is 3 too, and the least sum is S(3) = 3 + 3 = 6
Problems with a root function
For example: The distance from the point (x, x) on the line y = x to the point (2, 0) is D(x) = √((x − 2)² + x²) . For which value of x is the distance least?
Answer and solution
The answer: 1
Expand under the root: D(x) = √(2x² − 4x + 4) . The derivative: D′(x) = (2x − 2) / D(x) . The root in the denominator is positive, so the derivative has the sign of the numerator. The derivative is 0 at x = 1 . For x < 1 the derivative is negative, and for x > 1 it is positive. This is a minimum point. The distance is least at the point (1, 1) .
Problems with a box
For example: From a square sheet of side 6 , a square of side x is cut from each corner, and the edges are folded up into an open box. For which value of x is the volume of the box greatest?
Answer and solution
The answer: 1
The base is a square of side 6 − 2x , and the height is x . The volume: V(x) = x(6 − 2x)² V(x) = 4x³ − 24x² + 36x The domain: 0 < x < 3 V′(x) = 12x² − 48x + 36 . The derivative is 0 at x = 1 or at x = 3 , and only the first is in the domain. For x < 1 the derivative is positive, and for x > 1 it is negative. This is a maximum point. The volume is greatest when x = 1 .
Greatest and least values on a closed interval
For example: Here is the function f(x) = x + 1 / x . Find its greatest value on the interval 1 ≤ x ≤ 4 .
Answer and solution
The answer: 4.25
The derivative: f′(x) = 1 − 1 / x² . Inside the interval the derivative is positive, so the function increases on all of it. At the ends: f(1) = 2 and f(4) = 4 + 0.25 = 4.25 . The greatest value is 4.25 .
Not on the site yet
Parts of this topic that are not built yet: Extremum problems with triangles in a circle (solid geometry beyond the box).
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 4 units (grades 10 to 12).