Question

Discuss how to find the maximum or minimum function whose highest exponential function is 2. Discuss, if the highest exponential is 3.Discuss how to find the maximum or minimum function whose highest exponential function is 2. Discuss, if the highest exponential is 3.

Answer 1

Let us consider a function with highest exponential function as 2

f(x) = ax² + bx + c .................. (i)

to find maxima or minima, we take first derivative equal to zero

f ' (x) = 2ax + b = 0

x = -b/2a ................... (ii)

x < 0 implies minimum value.

Substituting, (ii) in (i) gives the minimum value of the function

f(x) = a (-b/2a) ² + b (-b/2a) + c

= ab²/ 4a² - b² /2a +c

= b² / 4a - b² /2a +c

=( b² - 2b² )/ 4a +c

f(x) = -b² /4a +c

The sign of a gives the maxima or minima, if a>0, then the function gives minimum value and if a< 0, gives maximum value.

In addition, the sign of the second derivative decides the maximua or minima

f '' (x) = 2a >0 implies minima.

Thus in the above function, x= -b/2a and the minimum value is -b² /4a +c. Hence, the parabola opens upward.

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Now, lets take the function with highest exponent function as 3

f(x) = -ax³ + bx² + c .................... (iii)

again to find maxima or minima, we take first derivative equal to zero

f ' (x) = -3ax² + 2bx = 0

= x(-3ax + 2b) = 0

x=0 or x = -2b/-3a = 2b/3a > 0

x is either 0 or 2b/3a.................. (iv)

substituting (iv) in (iii)

x=0

f(x) = -a(0)³ + b(0)² + c = c

x= 2b/3a

f(x) = -a(2b/3a)³ + b(2b/3a)² + c

= -a (8b³ / 27a³) + b(4b²/ 9a²) + c

= -8b³ /27a² + 4b³ /9a² + c

= (12 b³ - 8b³ ) / 27 a² + c

f(x) = 4 b³ / 27 a² + c

The sign of a gives the maxima or minima, if a>0, then the function gives minimum value and if a< 0, gives maximum value.

In the given function, a<0, the function has a maxima. Also, the second derivative,

f '' (x) = d(-3ax² + 2bx)/ dx

f '' (x) = -6ax + 2b

Thus in the above function, x= 0 or 2b/3a  and the maximum value is 4 b³ / 27 a² + c . Hence, the parabola opens downward.