Question

during which interval is the graph of the function below only increasing?
$f(x)=x^3 + 9x^2 + 24x + 15$
select one:
a. $-\infty < x < \infty$
b. $-\infty < x < -2$
c. $-5 < x < -2$
d. $-2 < x < \infty$
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what is a zero of the function below?
$f(x)=3^x - 243$
select one:
a. $x=-5$
b. $x=5$
c. $x=-9$
d. $x=9$

Explanation:

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First Question:

Step1: Find derivative of $f(x)$

$f'(x) = 3x^2 + 18x + 24$

Step2: Factor the derivative

$f'(x) = 3(x^2 + 6x + 8) = 3(x+2)(x+4)$

Step3: Find critical points

Set $f'(x)=0$: $3(x+2)(x+4)=0$ → $x=-4, x=-2$

Step4: Test sign of $f'(x)$

• For $x < -4$: $f'(x) > 0$ (function increasing)
• For $-4 < x < -2$: $f'(x) < 0$ (function decreasing)
• For $x > -2$: $f'(x) > 0$ (function increasing)

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Second Question:

Step1: Set $f(x)=0$ for zero

$3^x - 243 = 0$

Step2: Isolate the exponential term

$3^x = 243$

Step3: Rewrite 243 as power of 3

$3^x = 3^5$

Step4: Solve for $x$

Since bases are equal, $x=5$

Answer:

1. d. $-2 < x < \infty$
2. b. $x = 5$