Question

which parent functions have extrema?
square root
greatest integer
exponential
quadratic
linear
absolute value
cubic
rational

Explanation:

To determine which parent functions have extrema, we analyze each function:

1. Square Root: The parent square root function \( y = \sqrt{x} \) (domain \( x \geq 0 \)) is strictly increasing, so it has no extrema (no maximum or minimum in its domain, as it approaches \( 0 \) as \( x \to 0 \) and increases without bound as \( x \to \infty \)).
2. Greatest Integer: The greatest integer function (floor function \( y = \lfloor x

floor \)) is piecewise constant and non - differentiable at integers. It has no extrema (it does not have a single maximum or minimum over its domain \( \mathbb{R} \)).

1. Exponential: The parent exponential function \( y = a^x \) (\( a>0, a

eq1 \)) is either strictly increasing (\( a > 1 \)) or strictly decreasing (\( 0 < a < 1 \)), so it has no extrema.

1. Quadratic: The parent quadratic function \( y = x^2 \) has a parabola opening upward. Its vertex at \( (0,0) \) is a minimum (no maximum, as \( x \to \pm\infty \), \( y \to \infty \)). Thus, it has a minimum (extremum).
2. Linear: The parent linear function \( y = x \) is strictly increasing (slope \( 1 \)) and has no extrema (no maximum or minimum over \( \mathbb{R} \)).
3. Absolute Value: The parent absolute value function \( y = |x| \) has a V - shape with a vertex at \( (0,0) \), which is a minimum (as \( x \to \pm\infty \), \( y \to \infty \)). Thus, it has a minimum (extremum).
4. Cubic: The parent cubic function \( y = x^3 \) has a point of inflection at \( (0,0) \) and is strictly increasing (derivative \( y'=3x^2\geq0 \), with equality only at \( x = 0 \)), so it has no extrema.
5. Rational: The parent rational function (e.g., \( y=\frac{1}{x} \)) has no extrema over its domain (it has vertical asymptotes and is not bounded, so no maximum or minimum).

Answer:

Quadratic, Absolute Value