QuestionWhich function can have multiple increasing intervals? a. Square root b. Cube root c. Quadratic d. Cubic

Studdy Solution

STEP 1

Assumptions1. We are looking for a function family that can have more than one increasing interval. . An increasing interval of a function is an interval where the function values increase as the input values increase.
3. The function families we are considering are square root, cube root, quadratic, and cubic.

STEP 2

Let's analyze each function family.
a. Square root functions The general form is $f(x) = \sqrt{x}$ . These functions are always increasing, but they only have one increasing interval.
b. Cube root functions The general form is $f(x) = \sqrt[]{x}$ . These functions are always increasing, but they only have one increasing interval.
c. Quadratic functions The general form is $f(x) = ax^2 + bx + c$ . These functions are either always increasing or always decreasing, depending on the sign of $a$ . They do not have more than one increasing interval.

STEP 3

Now let's consider the cubic functions. The general form is $f(x) = ax^3 + bx^2 + cx + d$ . Depending on the values of $a$ , $b$ , $c$ , and $d$ , these functions can have more than one increasing interval.
For example, consider the cubic function $f(x) = x^3 -3x^2 -9x +27$ . This function increases from $-\infty$ to $-1$ , decreases from $-1$ to $3$ , and increases again from $3$ to $\infty$ .
Therefore, the correct answer is d. Cubic.

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