Question

he condition that f(x) = ax

3 + bx

2 + cx + d has no extreme value is

1) b

2

− 4ac = 0 2) b

2 = 3ac 3) b

2

< 3ac 4) b

2

> 3ac

Answer 1

Given: The equation f(x) = ax^3 + bx^2 + cx + d

To find: The condition that f(x) has no extreme value?

Solution:

• Now we have given the equation  f(x) = ax^3 + bx^2 + cx + d

• Now we will differentiate it with respect to x, we get:

                 f'(x) = 3ax^2 + 2bx + c

• Now the condition for no extremes is:

• f should have zero turning points, that is above quadratic equation should have no real roots.  

• So the condition is b^2 - 4(a)(c) < 0

                 (2b)^2 - 4(3a)(c) < 0

                 4b^2 - 12ac < 0

                 4b^2 < 12ac

                 b^2 < 3ac

Answer:

         So the condition that f(x) has no extreme value is b^2 < 3ac.