Question 1

Find all intervals where the derivative of the function shown below is negative.

Accepted Answer

2 < x < 3

Question 2

The derivative of function shown below is negative on the interval shown.

Accepted Answer

False

Question 3

Find all intervals where the derivative of the function shown below is negative.

Accepted Answer

-5 < x < 0 and 0 < x < 3

Question 4

Find the intervals of increase and decrease for the function $f ( x ) = x ^ { 2 } + 5 x - 3$ . A) $\text { Decreasing for } x < - \frac { 5 } { 2 } \text {; increasing for } x > - \frac { 5 } { 2 }$ B) $\text { Decreasing for } x > - \frac { 5 } { 2 } \text {; increasing for } x < - \frac { 5 } { 2 }$ C) Decreasing for all x D) Increasing for all x

Accepted Answer

To find intervals of increase and decrease, we first find the derivative of $f(x)$, which is $f'(x) = 2x + 5$. Setting $f'(x) = 0$ gives $x = -\frac{5}{2}$. This is a critical point. To the left of this point, the derivative is negative (indicating a decrease), and to the right, it is positive (indicating an increase). Therefore, the function is decreasing for $x < -\frac{5}{2}$ and increasing for $x > -\frac{5}{2}$.

Question 5

Find the intervals of increase and decrease for the function $f ( x ) = x ^ { 2 } + 9 x - 4$ . A) $\text { Decreasing for } x > - \frac { 9 } { 2 } \text { and increasing for } x < - \frac { 9 } { 2 }$ B) $\text { Decreasing for } x < - \frac { 9 } { 2 } \text { and increasing for } x > - \frac { 9 } { 2 }$ C) Decreasing for all x D) Increasing for all x

Accepted Answer

To determine the intervals of increase and decrease, we first find the derivative of $f(x)$, which is $f'(x) = 2x + 9$. Setting $f'(x) = 0$ gives $x = -\frac{9}{2}$. This is a critical point. To the left of this point ($x < -\frac{9}{2}$), $f'(x)$ is negative, indicating $f(x)$ is decreasing. To the right of this point ($x > -\frac{9}{2}$), $f'(x)$ is positive, indicating $f(x)$ is increasing.

Question 6

Find the intervals of increase and decrease for $f ( x ) = 6 x ^ { 3 } + 9 x ^ { 2 } - 108 x - 2$ . A) Increasing on x$\le$ -2 and x $\ge$ 3, decreasing on - 2 $\le$ x$\le$ 3 B) Increasing on x < -3 and x > 2, decreasing on -3 < x < 2 C) Increasing on -3 < x < 2, decreasing on x < -3 and x > 2 D) Increasing on x < -2, decreasing on x > 3

Accepted Answer

To determine the intervals of increase and decrease for the function $f(x) = 6x^3 + 9x^2 - 108x - 2$, we first find its derivative, $f'(x) = 18x^2 + 18x - 108$. Factoring this gives $f'(x) = 18(x^2 + x - 6) = 18(x + 3)(x - 2)$. Setting $f'(x) = 0$ gives the critical points $x = -3$ and $x = 2$. Testing values in the intervals $(-\infty, -3)$, $(-3, 2)$, and $(2, \infty)$ in the derivative $f'(x)$ will show that the function is increasing when $x < -3$ and $x > 2$, and decreasing when $-3 < x < 2$.

Question 7

Find the intervals of increase and decrease for the function $f ( x ) = - x ^ { 5 } - x ^ { 3 } - 4 x + 10$ .

Accepted Answer