Question

12 mark for review
graph of ( f )
the graph of ( f ), the derivative of the function ( f ), is shown above for ( -1 < x < 5 ). which of the following statements is true for ( -1 < x < 5 )?
a ( f ) has one relative minimum and two relative maxima.
b ( f ) has two relative minima and one relative maximum.
c ( f ) has two relative minima and three relative maxima.
d ( f ) has three relative minima and three relative maxima.

Explanation:

Step1: Recall Critical Point Rule

A function \( f(x) \) has a relative extremum (minimum or maximum) where \( f'(x) = 0 \) and \( f'(x) \) changes sign. A relative minimum occurs when \( f'(x) \) changes from negative to positive, and a relative maximum occurs when \( f'(x) \) changes from positive to negative.

Step2: Analyze \( f'(x) \) Sign Changes

- **First Critical Point (near \( x=0 \))**: \( f'(x) \) goes from negative (left of \( x=0 \)) to positive? Wait, no—wait, the graph at \( x=0 \): left of \( x=0 \), \( f'(x) \) is increasing from negative to 0 at \( x=0 \), then decreases. Wait, no, let's look at the zeros of \( f'(x) \): the graph crosses the x-axis at \( x=0 \), \( x=2 \), \( x=3 \), \( x=4 \)? Wait, no, the graph: at \( x=0 \), it touches the origin (maybe a zero with a tangent? Wait, no, the graph: let's list the zeros where \( f'(x) = 0 \) and sign changes.

Wait, the key is: relative minima of \( f \) occur where \( f' \) goes from - to +; relative maxima where \( f' \) goes from + to -.

Looking at the graph of \( f' \):

1. **First zero (x=0? Wait, no, left of x=0, f' is negative, at x=0, it's zero, then after x=0, f' goes negative? Wait, no, the graph: left of x=0, f' is increasing from -infinity to 0 at x=0, then decreases (goes negative) after x=0? Wait, no, the graph: at x=0, it's a peak? Wait, no, the graph starts from below, rises to (0,0), then dips down, then rises again. Wait, maybe better to count the number of times \( f' \) changes sign from - to + (minima) and + to - (maxima).

Wait, let's track the sign of \( f'(x) \):

- For \( x < 0 \): \( f'(x) < 0 \) (since the graph is below x-axis).
- At \( x=0 \): \( f'(x) = 0 \). Then, after \( x=0 \), \( f'(x) \) becomes negative (dips below), then later (after some point) becomes positive (rises above x-axis around x=1-2), then negative again (dips below at x=2-3), then positive (rises at x=3-4, then zero at x=4, then positive after x=4).

Wait, the zeros of \( f'(x) \) (where it crosses or touches x-axis) are at \( x=0 \), \( x=2 \), \( x=3 \), \( x=4 \)? Wait, no, the graph:

- At \( x=0 \): touches the origin (maybe a zero with a tangent, but sign change? Left of x=0: \( f'(x) < 0 \); right of x=0 (immediately): \( f'(x) < 0 \)? No, that can't be. Wait, maybe I misread. Wait, the graph: starts from the bottom left (x < -1, f' is negative), rises to (0,0), then dips down (so f' is negative after x=0), then rises again, crosses x-axis at some point (maybe x=1.5?), reaches a peak above x-axis, then crosses x-axis at x=2, dips below, then rises to x=3 (touching x-axis? No, x=3 is a peak on the x-axis? Wait, the graph at x=3: it's a point on the x-axis, then dips, then rises to x=4 (crosses x-axis at x=4), then rises.

Wait, the correct way: relative minima of \( f \) are where \( f' \) goes from - to + (so \( f' \) crosses x-axis from below to above). Relative maxima of \( f \) are where \( f' \) goes from + to - (crosses x-axis from above to below).

Let's count:

- **Relative Minima (f' from - to +)**:
1. At x=0? No, left of x=0: f' is -; right of x=0: f' is - (since it dips down). Wait, no, maybe the first minimum is when f' goes from - to +: let's see, after x=0, f' dips, then rises, crosses x-axis (from - to +) – that's one minimum. Then, later, f' goes from - to + again? Wait, no, let's look at the graph:

Wait, the graph of \( f' \):

- From x < 0: f' is negative, increasing to (0,0).
- Then, f' decreases (negative) until some point, then increases, crosses x-axis (from negative to positive) – that's a relative minimum of f (since f' goes from - to +).
- Then, f' increases to a peak (positive), then decreases, crosses x-axis (from positive to negative) – that's a relative maximum of f (f' goes from + to -).
- Then, f' decreases (negative), then increases, touches x-axis at x=3 (but sign doesn't change? Wait, x=3: f' is zero, but left of x=3: f' is negative, right of x=3: f' is negative? No, after x=3, f' dips, then rises, crosses x-axis at x=4 (from negative to positive) – that's another relative minimum of f.
- Then, f' increases (positive) after x=4.

Wait, no, let's list the sign changes:

1. **First sign change (from - to +)**: When f' goes from negative to positive. Looking at the graph, after x=0, f' dips, then rises, crosses x-axis (from - to +) – that's one minimum.
2. **Second sign change (from - to +)**: At x=4, f' crosses from negative to positive – that's another minimum. So two relative minima.

Now, relative maxima (f' from + to -):

1. **First sign change (from + to -)**: When f' goes from positive to negative. After the first minimum (crossing from - to +), f' rises to a peak (positive), then crosses x-axis from positive to negative – that's one maximum.
2. **Second sign change (from + to -)**: Wait, at x=3, is that a sign change? No, x=3: f' is zero, but left and right: f' is negative? Wait, no, maybe I missed. Wait, the graph: after x=2 (crossing from + to -), f' goes negative, then rises to x=3 (a peak on the x-axis? No, x=3 is a local maximum of f'? Wait, no, the graph of f' has its own extrema, but we care about sign changes of f' for f's extrema.

Wait, the key is: the number of times f' changes from - to + (minima of f) and + to - (maxima of f).

Looking at the graph:

- f' is negative, then crosses to positive (first minimum of f) – 1 minimum.
- f' is positive, then crosses to negative (first maximum of f) – 1 maximum.
- f' is negative, then crosses to positive (second minimum of f) – 2 minima.
- Wait, but at x=0, does f' change sign? Left of x=0: f' is negative; right of x=0: f' is negative (since it dips down). So no sign change at x=0. Then, after x=0, f' goes negative, then rises, crosses x-axis (from - to +) – that's first minimum. Then f' goes positive, rises, then falls, crosses x-axis (from + to -) – first maximum. Then f' goes negative, falls, then rises, touches x-axis at x=3 (but sign doesn't change, since left and right are negative? No, maybe x=3 is a zero with f' not changing sign (like a tangent), so not a critical point for f's extrema. Then f' continues negative, then rises, crosses x-axis at x=4 (from - to +) – second minimum. Then f' goes positive.

Wait, but also, at x=0, is that a critical point? f'(0)=0, but since f' doesn't change sign (left and right are negative), it's not a relative extremum of f.

Wait, maybe I made a mistake. Let's recall: the number of relative minima of f is the number of times f' goes from - to +, and relative maxima is the number of times f' goes from + to -.

Looking at the graph of f':

- From left (x < 0): f' is negative, increasing to (0,0). Then, f' decreases (so f' is negative after x=0), then increases, crosses x-axis (from negative to positive) – that's a - to + change (1 minimum).
- Then, f' increases to a peak (positive), then decreases, crosses x-axis (from positive to negative) – that's a + to - change (1 maximum).
- Then, f' decreases (negative), then increases, touches x-axis at x=3 (but f' is negative on both sides? No, maybe x=3 is a local maximum of f', but f' is negative there? Wait, no, the graph at x=3: it's a point on the x-axis, so f'(3)=0. Left of x=3: f' is negative (since after crossing from + to - at x=2, f' goes negative, then rises to x=3 (so left of x=3: f' is negative, right of x=3: f' is negative (dips again), then rises to x=4 (crosses from negative to positive) – so at x=3, f' doesn't change sign (both sides negative), so not an extremum of f.
- Then, f' rises from negative to positive at x=4 – that's a - to + change (2nd minimum).
- Wait, but also, at x=0, f' is zero, but left and right are negative, so no sign change. Then, is there another maximum? Wait, maybe the graph has a peak above x-axis between x=1 and x=2, then a dip, then a peak at x=3 (on x-axis), then a dip, then rise. Wait, maybe I miscounted the sign changes.

Wait, let's list all the zeros of f' where sign changes:

1. Zero 1: x=a (between 0 and 2) where f' goes from - to + (minimum of f).
2. Zero 2: x=b (between 2 and 3) where f' goes from + to - (maximum of f).
3. Zero 3: x=4 where f' goes from - to + (minimum of f). Wait, no, that's two minima. But wait, the graph at x=0: f'(0)=0, but sign doesn't change. Then, is there another maximum? Wait, maybe the first zero is x=0, but sign doesn't change. Then, after x=0, f' goes negative, then positive (crossing x-axis: - to +, minimum), then positive to negative (crossing x-axis: + to -, maximum), then negative to positive (crossing x-axis: - to +, minimum), and also, at x=3, f' is zero, but sign doesn't change. Wait, but the options:

Option B: two minima, one maximum. Wait, no, maybe I missed a maximum. Wait, the graph of f' has a peak at x=0? No, x=0 is a point where f' is zero, then it dips. Wait, maybe the correct count is:

- Relative minima of f: where f' changes from - to +. Looking at the graph, f' is negative, then becomes positive (first time: after x=0, crosses from - to +) – 1. Then, f' is negative again (after x=2), then becomes positive (at x=4, crosses from - to +) – 2. So two minima.

- Relative maxima of f: where f' changes from + to -. f' is positive, then becomes negative (after the first minimum, f' rises to positive, then falls to negative at x=2) – 1. Then, is there another? Wait, at x=3, f' is zero, but f' is negative on both sides? No, maybe the graph has a peak at x=3 (on x-axis), but f' is zero there, and left and right are negative, so no sign change. Wait, but the options:

Option B: two minima, one maximum. Wait, but let's check the options again.

Wait, the graph of f':

- The derivative f' crosses the x-axis at x=0, x=2, x=3, x=4? Wait, no, x=0: f'(0)=0 (touching), x=2: crosses from + to -, x=4: crosses from - to +. Wait, maybe x=0 is a zero with f' not changing sign (since left and right are negative), x=2: f' changes from + to -, x=4: f' changes from - to +. But also, between x=0 and x=2, f' goes from - to + (crossing x-axis at some point between 0 and 2), then from + to - (crossing at x=2), then from - to + (crossing at x=4). Wait, that would be two minima (crossing from - to + at two points: between 0-2 and at 4) and one maximum (crossing from + to - at x=2). But that's not matching. Wait, maybe the graph has three sign changes? No, let's look at the options:

Option B: two relative minima and one relative maximum.

Wait, let's recall: the number of relative minima of f is the number of times f' goes from - to +, and relative maxima is the number of times f' goes from + to -.

Looking at the graph:

- f' starts negative, increases to 0 at x=0 (no sign change, since left and right are negative).
- Then, f' decreases (negative), then increases, crosses x-axis from negative to positive (first - to +: 1 minimum).
- Then, f' increases to positive, then decreases, crosses x-axis from positive to negative (first + to -: 1 maximum).
- Then, f' decreases (negative), then increases, touches x-axis at x=3 (no sign change, since left and right are negative).
- Then, f' increases, crosses x-axis from negative to positive (second - to +: 2nd minimum).

Wait, but that's two minima and one maximum, which is option B. Wait, but the initial thought was maybe more, but no. Wait, the graph of f' has:

- Two points where f' goes from - to + (minima of f): so two minima.
- One point where f' goes from + to - (maximum of f): so one maximum.

Thus, option B is correct.

Answer:

B. \( f \) has two relative minima and one relative maximum.