Is #f(x)=-2x^5-3x^4+15x-4# concave or convex at #x=-4#?

Answer 1

Convex (sometimes called "concave upwards").

The concavity and convexity of a function can be determined by examining the sign of a function's second derivative.

Note that: you may call concave "concave down" and convex "concave up."

We must find the function's second derivative through the power rule:

#f(x)=-2x^5-3x^4+15x-4#
#f'(x)=-10x^4-12x^3+15#
#f''(x)=-40x^3-36x^2#
The value of the second derivative at #x=-4# is:
#f''(-4)=-40(-4)^3-36(-4)^2=1984#
Since this is #>0#, the function is convex (sometimes called concave up) at #x=-4#.
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To determine the concavity of a function at a specific point, we need to examine the second derivative of the function at that point. If the second derivative is positive, the function is concave upward (convex) at that point. If the second derivative is negative, the function is concave downward at that point. If the second derivative is zero, the test is inconclusive.

Given ( f(x) = -2x^5 - 3x^4 + 15x - 4 ), we need to find the second derivative and evaluate it at ( x = -4 ).

First, find the first derivative of ( f(x) ): [ f'(x) = -10x^4 - 12x^3 + 15 ]

Now, find the second derivative: [ f''(x) = -40x^3 - 36x^2 ]

Evaluate the second derivative at ( x = -4 ): [ f''(-4) = -40(-4)^3 - 36(-4)^2 = -40(-64) - 36(16) = 2560 - 576 = 1984 ]

Since ( f''(-4) = 1984 ) which is positive, the function is concave upward (convex) at ( x = -4 ).

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7