Absolute maximum and minimum of f(x) = -x^3 + 3x^2 on [-1, 3] Step by Step

Step by step

The question

Question: find the absolute maximum and minimum of f(x) = −x³ + 3x² on [−1, 3].

f(x) = −x³ + 3x²

On a closed interval, the highest and lowest values are at an end, or where the slope is 0.

So we test those, and only those.

candidates: the two ends + where f′ = 0

1 Critical points

Use the power rule on each term.

f′(x) = −3x^(3 − 1) + 3 × 2x^(2 − 1)

Now we have f′(x) = −3x² + 6x.

= −3x² + 6x

Critical points: where the slope is 0.

Solve f′(x) = 0.

f′(x) = 0

−3x² + 6x = 0

Take out the −3.

−3(x² − 2x) = 0

We need two numbers that multiply to 0 and add to −2: 0 and −2.

−3x(x − 2) = 0

0 × (−2) = 0, 0 + (−2) = −2

A product is 0 only when one of its parts is 0.

a × b = 0 → a = 0 or b = 0

So x = 0 or x = 2.

x = 0 or x = 2

2 Keep the ones inside

x = 0 is inside the interval.

Keep it.

x = 0 in [−1, 3] ✓

x = 2 is inside the interval.

Keep it.

x = 2 in [−1, 3] ✓

3 Test every candidate

The left end: f of −1 is 4.

f(−1) = −(−1)³ + 3(−1)² = 1 + 3 = 4

The critical point: f of 0 is 0.

f(0) = −(0)³ + 3(0)² = 0 + 0 = 0

The critical point: f of 2 is 4.

f(2) = −2³ + 3(2)² = −8 + 12 = 4

The right end: f of 3 is 0.

f(3) = −3³ + 3(3)² = −27 + 27 = 0

4 Biggest and smallest

Careful!

The maximum is at an end, where the slope is not 0.

That is why we test the ends.

Careful: the ends count too

The biggest value is 4, at x = −1 and x = 2.

That is the absolute maximum.

max : 4 at x = −1 and x = 2

The smallest value is 0, at x = 0 and x = 3.

That is the absolute minimum.

min : 0 at x = 0 and x = 3

5 Check

Let's check with the graph, only from −1 to 3.

The highest point is at x = −1 and x = 2, and the lowest at x = 0 and x = 3.

Right!

Let's look back.

First: critical points.

Then: keep the ones inside.

Then: test every candidate.

Then: biggest and smallest.

Last: check.

Answer