Find the value of x at which the following function has a local minimum:
𝑓
(
𝑥
)
=
𝑥
(
𝑥
2
−
5
)
1
3

Question

Find the value of x at which the following function has a local minimum: 
𝑓
(
𝑥
)
=
𝑥
(
𝑥
2
−
5
)
1
3

BlackTom AI · Accepted Answer

The problem asks for the value(s) of x where the function f(x) = x (x^2 − 5)^{1/3} has a local minimum. I’ll start by differentiating the function carefully. Let g(x) = (x^2 − 5)^{1/3}. Then f(x) = x g(x). Using the product rule, f'(x) = g(x) + x g'(x). Next, compute g'(x): g'(x) = (1/3)(x^2 − 5)^{−2/3} · (2x) = (2x/3)(x^2 − 5)^{−2/3}. Substituting back gives f'(x) = (x^2 − 5)^{1/3} + x · (2x/3)(x^2 − 5)^{−2/3}. To combine terms, factor out (x^2 − 5)^{−2/3}: f'(x) = (x^2 − 5)^{−2/3} [ (x^2 − 5) + (2x^2)/3 ]. This simplifies to f'(x) = (1/3)(x^2 − 5)^{−2/3} [3x^2 − 15 + 2x^2] = (1/3)(x^2 − 5)^{−2/3} (5x^2 − 15). Thus f'(x) = (5/3)(x^2 − 5)^{−2/3} (x^2 − 3). The critical points occur when f'(x) = 0 or when f'(x) is undefined. The factor (x^2 − 3) = 0 gives x = ±√3. The derivative is undefined where x^2 − 5 = 0, i.e., x = ±√5, since (x^2 − 5)^{−2/3} blows up there. Between these points, the sign of f'(x) is governed by the sign of (x^2 − 3) because the remaining factor (x^2 − 5)^{−2/3} is positive for all x ≠ ±√5 (the −2/3 exponent yields a positive magnitude regardless of the sign of x^2 − 5). Therefore, the sign pattern of f'(x) is determined by x^2 − 3: - If |x| < √3, then x^2 − 3 < 0, so f'(x) < 0 (decreasing). - If √3 < |x| < √5, then x^2 − 3 > 0, so f'(x) > 0 (increasing). - If |x| > √5, then x^2 − 3 > 0, so f'(x) > 0 (still increasing). Consequently, f'(x) changes sign from negative to positive as x passes through x = √3 (from left to right, f' goes negative to positive), indicating a local minimum there. At x = -√3, f'(x) changes from positive to negative, indicating a local maximum. The points x = ±√5 are locations where the derivative is not defined (cusps) but do not involve a change in the monotonicity direction, so they do not correspond to local extrema. In summary, the local minimum occurs at x = √3, with the surrounding sign analysis showing the transition from decreasing to increasing at that point. The value of x where this occurs is therefore √3 (approximately 1.732...). To cross-check, one can evaluate f'(x) on intervals and observe the sign changes described above, which align with the identified critical points and their nature.

ECON6003/ECON6703 Mathematical Methods of Econ Analysis Assignment 1

Find the value of x at which the following function has a local minimum: 𝑓 ( 𝑥 ) = 𝑥 ( 𝑥 2 − 5 ) 1 3

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