Expert Answer to "Concave or convex at x=−1?f(x)=x4−4x3+x−4"

Nancy Richmond

Answered question

2022-03-25

Concave or convex at

x = - 1

f (x) = x^{4} - 4 x^{3} + x - 4

SofZookywookeoybd

Beginner2022-03-26Added 15 answers

Step 1
To find that out, we need to get the second derivative first.
Getting the first derivative.
1)

f^{'} (x) = \frac{d}{d x} (x^{4} - 4 z^{3} + x - 4)

We can easily find this using power rule.
2)

f^{'} (x) = 4 x^{3} - 12 x^{2} + 1

Getting the second derivative.
1)

f^{″} (x) = \frac{d}{d x} (4 x^{3} - 12 x^{2} + 1)

Use power rule again.
2)

f^{″} (x) = 12 x^{2} - 24 x

Now that we know the second derivative, we will evaluate

f^{″} (x)

x = - 1

to check its concavity.
If

f^{″} (x) > 0

, then it is concave up or convex
If

f^{″} (x) < 0

, then it is concave down or concave
1)

f^{″} (- 1) = 12 {(- 1)}^{2} - 24 (- 1)

f^{″} (- 1) = 12 + 24

f^{″} (- 1) = 36

Since

f (x)

is 36 at

x = - 1

and 36 is greater than 0, then

f (x)

is convex at

x = - 1

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