Question

f(x)=cos(x), f(x)=|x|, f(x)=x^4

Answer 1

Do you want derivatives, integrals, plots, continuity/differentiability, extrema, Taylor series, or something else for these functions? I can do any of those — tell me which.

Below is a short useful summary for each; tell me which parts to expand.

1) f(x) = cos x
- Domain: all real x.
- Range: [-1, 1].
- Continuity/differentiability: smooth (infinitely differentiable).
- Derivative: f'(x) = -sin x. Second derivative: f''(x) = -cos x.
- Critical points: f'(x)=0 at x = nπ (n ∈ Z). At x = 2kπ cos = 1 (local maxima); at x = (2k+1)π cos = -1 (local minima).
- Integral: ∫cos x dx = sin x + C.
- Periodic with period 2π.

2) f(x) = |x|
- Domain: all real x.
- Range: [0, ∞).
- Continuity/differentiability: continuous everywhere; not differentiable at x = 0 (corner). Differentiable for x ≠ 0.
- Derivative: f'(x) = 1 for x > 0, f'(x) = -1 for x < 0; no derivative at 0. Can be written as sgn(x) for x ≠ 0.
- Critical point: x = 0 is the global minimum (value 0).
- Integral (antiderivative): ∫|x| dx = { x^2/2 + C for x ≥ 0; -x^2/2 + C for x ≤ 0 } (consistent up to constant).
- Convex: yes (|x| is convex on R).

3) f(x) = x^4
- Domain: all real x.
- Range: [0, ∞).
- Continuity/differentiability: smooth (infinitely differentiable).
- Derivative: f'(x) = 4x^3. Second derivative: f''(x) = 12x^2 ≥ 0, so convex on R.
- Critical points: f'(x)=0 at x = 0 (global minimum, value 0). It's a flat minimum (higher-order zero derivative).
- Integral: ∫x^4 dx = x^5/5 + C.

If you want plots, Taylor expansions, solve equations like f(x)=a, or more detailed analysis (e.g., behavior at infinity, Lipschitz/Holder, arc length), tell me which function(s) and what work you want.