A differentiable function f(x) has a relative minimum at x=0, then the function y=f(x)+a x+b has

step 1 We are given that a function f x Fx has a relative relative minima minimum minimum at x equal to

A differentiable function f(x) has a relative minimum at...

A differentiable function $\mathrm{f}(\mathrm{x})$ has a relative minimum at $x=0$, then the function $y=f(x)+a x+b$ has a relative minimum at $x=0$ for
(A) all a and all $b$
(B) all b if $\mathbf{a}=0$
(C) all $b>0$
(D) all $\mathrm{a}>0$

Key Concepts

Relative Minimum

A relative minimum of a differentiable function is a point where the function attains a value lower than at nearby points. At a relative minimum occurring at a point x = 0, the first derivative is zero (f'(0) = 0), and the second derivative, if it exists, is positive. These conditions are used to identify and verify the presence of a local minimum.

Impact of Adding a Linear Term to a Function

Adding a linear term ax to a function shifts its derivative by a constant a. This is because the derivative of ax is just a. Therefore, for a function f(x) that has a derivative of zero at x = 0, the modified function f(x) + ax + b will have a derivative of a at x = 0. To maintain a critical point at x = 0, which is necessary for a relative minimum, the additional term must not alter the zero derivative condition; hence a must equal zero. The constant b only shifts the function vertically and does not affect the location of the minimum.

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