Paper 4: Fundamentals of Business Mathematics & Statistic

3.54 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS

Calculus

4. Show that f(x) = x^3 – 3x^2 + 6x + 3 has neither maximum nor a minimum.


5. Show that the function y = x–3 – 3x^2 + 5 has a maximum value at x = 0 and a minimum value at x = 2.


6. Show that the fuction x^3 – 6x^2 + 12x + 50 is either a maximum nor a minimum at x = 2.


7. Find for what values of x, following expression is maximum and minimum respectively
   2x^3 – 21x^2 + 36x – 20. Find also the maximum and minimum values.
   [Ans. minimum at x = 6; maximum at x = 1; minimum value = –128; maximum value = –3]


8. Show that the function x (1–x^2 ) attains the maximum value at x =

1

3 and minimum value at

x^1

3

= −

9. show that y = x–3 – 8 has neither a maximum nor a minimum value. Has the curve a point of inflexion.
   [Ans. Yes at 0, –8]


10. Show that the function f(x) x=^2 +^250 x minimum value at x = 5.


11. Show that the function f(x) = x–3 – 6x^2 + 9x – 8 has a maximum value at x = 1 and a minimum value at
    x = 3.


12. (i) A steel plant produces x tons of steel per week at s total cost of

(^1) x 7x 11x 50. 3 2
3
(^) − + +
` (^)
Find the output level at which the marginal cost attains its minimum (using the concept of derivative
as used in finding extreme values). [Ans. 7]
(ii) A firm produces x tons valuable metal per month ata a total cost c given by
c^1 x 5x 75x 10.^32
3

=^ − + +^

` (^)
Find at what level of output the marginal cost attains its minimum. [Ans. 5]

13. The total cost of output x given by c x=2 353 2+
    Find: (i) cost when output is 4 units. (ii) average cost of output of 10 units.

(iii) marginal cost when output is 3 units. [Ans. 20 ,2 ;1 5 26 12 3]

14. The demand function faced by a afirm is p = 500 – 0.2x and its cost function is c = 25x + 10000
    (p = price, x = output and c = cost). Find the output at which the profits of the firm are maximum.

Also find the price it will charge. [Ans. 1187 ; 262.50^12 ` ]