a. Let f(x) be a function satisfying |f(x)| ≤ x² for −1 ≤ x ≤ 1. Show that f is differentiable at x = 0 and find f′(0).

Verified step by step guidance

1. To show that f is differentiable at x = 0, we need to check if the limit \( \lim_{h \to 0} \frac{f(h) - f(0)}{h} \) exists. Since |f(x)| ≤ x², we know that |f(h)| ≤ h² for small values of h.
2. Evaluate the expression \( \frac{f(h) - f(0)}{h} \). Since f(0) is bounded by 0², we have |f(0)| ≤ 0, which implies f(0) = 0.
3. Substitute f(0) = 0 into the limit expression: \( \lim_{h \to 0} \frac{f(h)}{h} \). Given |f(h)| ≤ h², we have \( -h^2 \/ ≤ f(h) \/ ≤ h^2 \).
4. Divide the inequality by h (assuming h ≠ 0): \( -h \/ ≤ \frac{f(h)}{h} \/ ≤ h \). As h approaches 0, both -h and h approach 0.
5. By the Squeeze Theorem, \( \lim_{h \to 0} \frac{f(h)}{h} = 0 \). Therefore, f is differentiable at x = 0 and f′(0) = 0.

Key Concepts

Differentiability

A function is differentiable at a point if it has a defined derivative at that point, which means the limit of the difference quotient exists. For a function f(x) to be differentiable at x = 0, the limit of (f(x) - f(0)) / (x - 0) as x approaches 0 must exist. Differentiability implies continuity, so if a function is differentiable at a point, it must also be continuous there.

Limit Definition of Derivative

The derivative of a function at a point can be defined using the limit: f'(a) = lim (h -> 0) [f(a + h) - f(a)] / h. In this case, to find f'(0), we need to evaluate the limit as h approaches 0 of [f(h) - f(0)] / h. This definition is crucial for determining the slope of the tangent line to the function at that point.

Bounded Functions

The condition |f(x)| ≤ x² for −1 ≤ x ≤ 1 indicates that the function f(x) is bounded by the parabola x² within the specified interval. This constraint helps in analyzing the behavior of f(x) as x approaches 0, particularly in ensuring that f(x) approaches 0 faster than x does, which is essential for proving differentiability at that point.