Math / Calculus

QuestionLet $h$ be a polynomial function and let $h^{\prime}$ , its derivative, be defined as $h^{\prime}(x)=x^{2}(x-2)^{2}(x-1)^{2}$ .
At how many points does the graph of $h$ have a relative maximum ? Choose 1 answer: (A) None (B) One (C) Two (D) Three

Studdy Solution

STEP 1

What is this asking? Where does the polynomial $h(x)$ have a hilltop, given its slope formula $h^{\prime}(x)$ ? Watch out! Don't mix up maximums and minimums!
We're looking for hilltops, not valleys.
Also, remember a critical point doesn't have to be a min or max!

STEP 2

1. Find the critical points
2. Analyze the sign of the derivative

STEP 3

To find the critical points, we need to figure out where the slope, given by $h^{\prime}(x)$ , is zero.
So, we set $h^{\prime}(x) = 0$ : $x^{2}(x-2)^{2}(x-1)^{2} = 0$

STEP 4

This equation is already nicely factored for us!
A product is zero if any of its factors are zero.
So, our critical points are $x = 0$ , $x = 2$ , and $x = 1$ .
Awesome!

STEP 5

The sign of the derivative tells us whether $h(x)$ is increasing (going uphill, positive slope) or decreasing (going downhill, negative slope).
A relative maximum happens when $h(x)$ switches from increasing to decreasing.

STEP 6

Look closely at $h^{\prime}(x) = x^{2}(x-2)^{2}(x-1)^{2}$ .
Everything is squared!
This means $h^{\prime}(x)$ will always be greater than or equal to zero.
It can be zero at our critical points, but everywhere else, it's positive.

STEP 7

Since $h^{\prime}(x)$ is always non-negative, $h(x)$ is always increasing or flat.
It never decreases!
Imagine a car that's always accelerating or coasting at a constant speed; it never brakes.
It's not going to have any hilltops!

STEP 8

Since $h(x)$ never switches from increasing to decreasing, it has no relative maximums.
The answer is (A) None.

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QuestionLet $h$ be a polynomial function and let $h^{\prime}$ , its derivative, be defined as $h^{\prime}(x)=x^{2}(x-2)^{2}(x-1)^{2}$ .
At how many points does the graph of $h$ have a relative maximum ? Choose 1 answer: (A) None (B) One (C) Two (D) Three

Studdy Solution

STEP 1

What is this asking? Where does the polynomial $h(x)$ have a hilltop, given its slope formula $h^{\prime}(x)$ ? Watch out! Don't mix up maximums and minimums!
We're looking for hilltops, not valleys.
Also, remember a critical point doesn't have to be a min or max!

STEP 2

1. Find the critical points
2. Analyze the sign of the derivative

STEP 3

To find the critical points, we need to figure out where the slope, given by $h^{\prime}(x)$ , is zero.
So, we set $h^{\prime}(x) = 0$ : $x^{2}(x-2)^{2}(x-1)^{2} = 0$

STEP 4

This equation is already nicely factored for us!
A product is zero if any of its factors are zero.
So, our critical points are $x = 0$ , $x = 2$ , and $x = 1$ .
Awesome!

STEP 5

The sign of the derivative tells us whether $h(x)$ is increasing (going uphill, positive slope) or decreasing (going downhill, negative slope).
A relative maximum happens when $h(x)$ switches from increasing to decreasing.

STEP 6

Look closely at $h^{\prime}(x) = x^{2}(x-2)^{2}(x-1)^{2}$ .
Everything is squared!
This means $h^{\prime}(x)$ will always be greater than or equal to zero.
It can be zero at our critical points, but everywhere else, it's positive.

STEP 7

Since $h^{\prime}(x)$ is always non-negative, $h(x)$ is always increasing or flat.
It never decreases!
Imagine a car that's always accelerating or coasting at a constant speed; it never brakes.
It's not going to have any hilltops!

STEP 8

Since $h(x)$ never switches from increasing to decreasing, it has no relative maximums.
The answer is (A) None.

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Studdy Solution

STEP 1

What is this asking? Where does the polynomial h(x)h(x)h(x) have a hilltop, given its slope formula h′(x)h^{\prime}(x)h′(x)? Watch out! Don't mix up maximums and minimums!We're looking for hilltops, not valleys.Also, remember a critical point doesn't *have* to be a min or max!

STEP 2

1. Find the critical points2. Analyze the sign of the derivative

STEP 3

To find the critical points, we need to figure out where the slope, given by h′(x)h^{\prime}(x)h′(x), is zero.So, we set h′(x)=0h^{\prime}(x) = 0h′(x)=0: x2(x−2)2(x−1)2=0x^{2}(x-2)^{2}(x-1)^{2} = 0x2(x−2)2(x−1)2=0

STEP 4

This equation is already nicely factored for us!A product is zero if any of its factors are zero.So, our critical points are x=0x = 0x=0, x=2x = 2x=2, and x=1x = 1x=1.Awesome!

STEP 5

The sign of the derivative tells us whether h(x)h(x)h(x) is increasing (going uphill, positive slope) or decreasing (going downhill, negative slope).A relative maximum happens when h(x)h(x)h(x) switches from increasing to decreasing.

STEP 6

Look closely at h′(x)=x2(x−2)2(x−1)2h^{\prime}(x) = x^{2}(x-2)^{2}(x-1)^{2}h′(x)=x2(x−2)2(x−1)2.Everything is squared!This means h′(x)h^{\prime}(x)h′(x) will *always* be greater than or equal to zero.It can be zero at our critical points, but everywhere else, it's positive.

STEP 7

Since h′(x)h^{\prime}(x)h′(x) is always non-negative, h(x)h(x)h(x) is always increasing or flat.It never decreases!Imagine a car that's always accelerating or coasting at a constant speed; it never brakes.It's not going to have any hilltops!

STEP 8

Since h(x)h(x)h(x) never switches from increasing to decreasing, it has *no* relative maximums.The answer is (A) None.

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

What is this asking? Where does the polynomial $h(x)$ have a hilltop, given its slope formula $h^{\prime}(x)$ ? Watch out! Don't mix up maximums and minimums!
We're looking for hilltops, not valleys.
Also, remember a critical point doesn't have to be a min or max!

1. Find the critical points
2. Analyze the sign of the derivative

To find the critical points, we need to figure out where the slope, given by $h^{\prime}(x)$ , is zero.
So, we set $h^{\prime}(x) = 0$ : $x^{2}(x-2)^{2}(x-1)^{2} = 0$

This equation is already nicely factored for us!
A product is zero if any of its factors are zero.
So, our critical points are $x = 0$ , $x = 2$ , and $x = 1$ .
Awesome!

The sign of the derivative tells us whether $h(x)$ is increasing (going uphill, positive slope) or decreasing (going downhill, negative slope).
A relative maximum happens when $h(x)$ switches from increasing to decreasing.

Look closely at $h^{\prime}(x) = x^{2}(x-2)^{2}(x-1)^{2}$ .
Everything is squared!
This means $h^{\prime}(x)$ will always be greater than or equal to zero.
It can be zero at our critical points, but everywhere else, it's positive.

Since $h^{\prime}(x)$ is always non-negative, $h(x)$ is always increasing or flat.
It never decreases!
Imagine a car that's always accelerating or coasting at a constant speed; it never brakes.
It's not going to have any hilltops!

Since $h(x)$ never switches from increasing to decreasing, it has no relative maximums.
The answer is (A) None.