Question

match each piece of derivative information with what it tells us about the graph of ( f(x) )

• ( f(x) ) is positive
• ( f(x) ) is negative
• ( f(x) ) is positive
• ( f(x) ) is negative

a. the graph of ( f(x) ) has an inflection point
b. the graph of ( f(x) ) is increasing
c. the graph of ( f(x) ) is concave down
d. the graph of ( f(x) ) is concave up
e. the graph of ( f(x) ) has a maximum or minimum
f. the graph of ( f(x) ) is decreasing
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question 2
given ( f(x) = 2x^3 + 12x^2 + 5x + 3 )
find where ( f(x) = 0 )
answer: ( x = ) (round to two decimal

Explanation:

First Matching Problem (Answer-Explanation Format)

• Positive first derivative means the function's output rises as input increases, so the graph is increasing.
• Negative first derivative means the function's output falls as input increases, so the graph is decreasing.
• Positive second derivative indicates the function's rate of change is increasing, so the graph curves upward (concave up).
• Negative second derivative indicates the function's rate of change is decreasing, so the graph curves downward (concave down).

Step1: Find first derivative $f'(x)$

Apply power rule: $\frac{d}{dx}[x^n]=nx^{n-1}$
$f'(x) = \frac{d}{dx}(2x^3+12x^2+5x+3) = 6x^2 + 24x + 5$

Step2: Find second derivative $f''(x)$

Differentiate $f'(x)$ with power rule
$f''(x) = \frac{d}{dx}(6x^2+24x+5) = 12x + 24$

Step3: Solve $f''(x)=0$

Set the second derivative equal to 0 and isolate $x$
$12x + 24 = 0$
$12x = -24$
$x = \frac{-24}{12} = -2.00$

Answer:

• $f'(x)$ is positive → b. The graph of $f(x)$ is increasing
• $f'(x)$ is negative → f. The graph of $f(x)$ is decreasing
• $f''(x)$ is positive → d. The graph of $f(x)$ is concave up
• $f''(x)$ is negative → c. The graph of $f(x)$ is concave down

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Second Calculus Problem (Step-by-Step Format)