If , then x = 0 is a point of:

Concept:

Continuity of a Function:

• A function f(x) is said to be continuous at a point x = a in its domain, if  exists or or if its graph is a single unbroken curve at that point.
• f(x) is continuous at x = a ⇔ .

Differentiability of a Function:

• A function f(x) is differentiable at a point x = a in its domain if its derivative is continuous at a.

  This means that f'(a) must exist, or equivalently: .

Maxima/Minima:

• If f(x) has a local maximum or a local minimum at a point x = a, then it must be either a critical point [f'(a) = 0] or a point of non-differentiability.

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Calculation:

Let us check for the continuity and differentiability (maxima/minima) of the function at x = 0.

Continuity:

 = 0² = 0.

 = 2 sin 0 = 0.

f(0) = 0² = 0.

∵  = f(0), the function f(x) is continuous at x = 0.

Differentiability:

 = 2×0 = 0.

 = 2 cos 0 = 2.

∵ , the function is not differentiable at x = 0.

Since, the function is not differentiable at x = 0, let us examine the possibility of maximum/minimum at the point.

The function f(x) = x² is strictly decreasing in (-∞, 0] and its minimum is 0² = 0 at x = 0.

The function f(x) = 2sin x is strictly increasing in  and its minimum is 2 sin 0 = 0 at x = 0.

∴ The function f(x) has a local minimum at x = 0.