How to Find the Critical Points and Inflection Points of the Function f(x) = x^5 – 70x^3 + 13?

To find the critical points and inflection points of the function f(x) = x^5 – 70x^3 + 13, we need to use derivatives.

Step 1: Find the first derivative

The first derivative, f'(x), helps us locate the critical points, where the function could have maxima, minima, or horizontal tangents:

f'(x) = 5x^4 - 210x^2

Step 2: Set the first derivative to zero

Next, we find where the first derivative is equal to zero:

5x^4 - 210x^2 = 0

Factoring out the common terms:

5x^2(x^2 - 42) = 0

This yields:

x^2 = 0  or  x^2 - 42 = 0

So, we find:

x = 0  or  x = ±√42

Step 3: Find the second derivative

The second derivative, f”(x), is necessary for determining the inflection points:

f''(x) = 20x^3 - 420x

Step 4: Set the second derivative to zero

To locate the inflection points, we set the second derivative to zero:

20x^3 - 420x = 0

Factoring out the common terms gives:

20x(x^2 - 21) = 0

This results in:

x = 0  or  x^2 - 21 = 0

So, we find:

x = 0  or  x = ±√21

Summary

In summary, the critical points of the function occur at:

• x = 0
• x = √42
• x = -√42

And the inflection points occur at:

• x = 0
• x = √21
• x = -√21