The range of values of k for which the function f(x) = (k2 - 4)x2 + 6x3 + 8x4 has a local maxima at point x = 0 is

The range of values of k for which the function f(x) = (k2 - 4)x2 + 6x3 + 8x4 has a local maxima at point x = 0 is Correct Answer -2 < k < 2

Concept:

Consider a function y = f(x) on a defined interval of x.

The function attains extreme values (the value can be maximum or minimum or both).

For maxima:

• Local maxima: A point is the local maxima of a function if there is some other point where the maximum value is greater than the local maxima but that point doesn’t exist nearby the local maxima.
• Global maxima: It is the point where there is no other point has in the domain for which function has more value than global maxima.

Condition:

f^"(x) < 0 ⇒ maxima

f"(x) > 0 ⇒ minima

f"(x) = 0 ⇒ Point of inflection

Calculation:

Given:

f(x) = (k2 - 4)x2 + 6x3 + 8x4 

f'(x) = 2(k2 - 4)x + 18x² + 32x³ 

f''(x) = 2(k2 - 4) + 36x + 96x² 

Since, at x = 0, f(x) has local maxima

f''(0) < 0

2(k2 - 4) + 36 × 0 + 96 × 0 < 0

k2 - 4 < 0

Here, to keep the above expression less than 0, the value of k must lie in between -2 to 2.

⇒ -2 < k < 2

Mistake Points

Since the condition for maxima is inequality, don't use it as an equation, i.e. k2 - 4 = 0. This will give k = ± 2 and changes answer to K < -2 or k > 2