How can we optimise for a non-zero value
𝑔
∗
of a particular function
𝑔
(
𝑥
)
?

Question

How can we optimise for a non-zero value  
𝑔
∗
of a particular function 
𝑔
(
𝑥
)
?

BlackTom AI · Accepted Answer

The question asks: How can we optimise for a non-zero value g* of a particular function g(x)?
Option 1: Define a new function f(x) = g(x) − g* and use a root finding method. This is a valid and common approach: if we want x such that g(x) = g*, we can subtract g* from g and seek the root of f(x) = 0. Employing a root-finding algorithm (e.g., bisection, Newton-Raphson) on f(x) will locate the x where g(x) equals the target g*, provided a solution exists and among the domain where f is continuous. This option correctly reframes the optimization task into a zero-finding problem, which is a standard technique for equality-constrained targets.

Option 2: First check if g* is less than g(0) to determine bounds, and then brute force search for when the derivative g′(x) is zero for x > 0 or x < 0. This option mischaracterizes the goal: optimizing for a non-zero target value g* does not require searching for points where the derivative is zero (those are stationary points, relevant for optimization criteria like maxima/minima of g, not specifically matching a value g*). Additionally, checking only g(0) for bounds is insufficient in general, and a brute-force search for a zero derivative does not guarantee achieving g(x) = g*. It conflates local extrema with exact target matching.

Option 3: It is not possible to optimise for a particular value. This is incorrect. In most mathematical settings, you can target or approximate a specific value of a function by transforming the problem (e.g., solving g(x) = g*) or by optimization techniques that minimize the squared error (g(x) − g*)^2. Declaring it impossible ignores standard methods for value-targeted optimization.

Option 4: Directly employ Newton's Method on the function g(x). Newton's Method is a root-finding technique, not an optimization method per se, and applying it directly to g(x) aims to find a root of g(x) = 0 rather than a generic target g(x) = g*. If you rewrite the problem as finding the root of f(x) = g(x) − g*, then Newton's Method can be used on f(x). However, applying Newton's Method directly to g(x) without adjusting for the target value does not solve the stated goal, so this option is incomplete/incorrect as stated.

In summary, the most accurate approach among the options is the first one, which converts the target value problem into a root-finding problem by subtracting g* and seeking f(x) = 0. The other choices either misinterpret the role of extrema, rely on insufficient bound handling, claim impossibility without justification, or misuse Newton's Method without adapting to the target value.

How can we optimise for a non-zero value 𝑔 ∗ of a particular function 𝑔 ( 𝑥 ) ?Single choice

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