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Consider function f(x) = (x2 - 4)2, where x is a real number. The function f(x) has

Consider function f(x) = (x2 - 4)2, where x is a real number. The function f(x) has Correct Answer Only two minima

Concept:

A point c in the domain of a function f at which either f’(c) = 0 or if f is no differentiable is called the critical point.

The slope of tangents at critical points is zero.

Let f be a real-valued function and c be an interior point in the domain of f such that f’(c) = 0

a) If f’(x) > 0 at every point close to the left of c and f’(x) < 0 at every point right of c, then it is a local maximum

b) If f’(x) > 0 at every point close to the right of c and f’(x) < 0 at every point left of c, then it is a local minimum

c) Also, if f’’(x) < 0 then it is maxima and if f’’(x) > 0 then it is minima.

d) If f’(x) does not change sign as x increases or decreases then such a point is called inflection.

Calculation:

f(x) = (x2 - 4)2

f’(x) = 2(x2 - 4)(2x) = 4x3 – 16x

Let f’(x) = 0

Then, x = 0, +2, -2

f’’(x) = 12x2 – 16

f’’(0) = -16 < 0 (maxima)

f’’(2) = 82 > 0 (minima)

f’’(-2) = 32 > 0 (minima)

Therefore, it has only two minima

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