Is #f(x) = -3x^4+x^3 - 4x^2+x-4# concave or convex at #x=11#?

Question

Isaiah Allen · Accepted Answer

The answer lies in the sign of the second derivative: the function is convex if the second derivative is positive, and concave otherwise. So, let's compute it. #f'(x) = -12x^3+3x^2-8x+1# #f''(x) = -36x^2+6x-8# Now evaluate #f''(11)#: #f''(11)=-36*(11^2) + 6*11 - 8 = -36*121 +66-8#, which is clearly negative (it's #-4298# to be precise), so the function is concave in that point.

Paisley Johnson · Answer

undefined To determine whether the function $f(x) = -3x^4 + x^3 - 4x^2 + x - 4$ is concave or convex at $x = 11$, we need to examine the second derivative of the function at that point.

First, we find the first and second derivatives of the function $f(x)$:
$$f'(x) = -12x^3 + 3x^2 - 8x + 1$$
$$f''(x) = -36x^2 + 6x - 8$$

Now, we evaluate $f''(11)$ to determine the concavity/convexity at $x = 11$:
$$f''(11) = -36(11)^2 + 6(11) - 8$$

After computing this value, we can conclude: 
- If $f''(11) > 0$, then $f(x)$ is convex at $x = 11$.
- If $f''(11) < 0$, then $f(x)$ is concave at $x = 11$.
- If $f''(11) = 0$, the test is inconclusive, and we may need to apply higher-order tests or examine the behavior around the point more closely.

Substitute the value of $x = 11$ into $f''(x)$, calculate the result, and determine whether it's positive or negative to ascertain whether $f(x)$ is concave or convex at $x = 11$.

HIX Tutor · Answer

undefined That's a great question! To find out if the function is concave or convex at $ x = 11 $, we can use the second derivative. Let's go step by step:

1. **Find the first derivative** $ f'(x) $:
   - Differentiate $ f(x) = -3x^4 + x^3 - 4x^2 + x - 4 $.
   - What do you think is the derivative of the terms?
  
2. **Calculate the first derivative**:
   - For $ -3x^4 $, the derivative is $ -12x^3 $.
   - For $ x^3 $, the derivative is $ 3x^2 $.
   - For $ -4x^2 $, the derivative is $ -8x $.
   - For $ x $, the derivative is $ 1 $.
   - So, $ f'(x) = -12x^3 + 3x^2 - 8x + 1 $.

3. **Find the second derivative** $ f''(x) $:
   - Differentiate $ f'(x) $.
   - Can you find the derivatives of the terms in $ f'(x) $?

4. **Calculate the second derivative**:
   - For $ -12x^3 $, the derivative is $ -36x^2 $.
   - For $ 3x^2 $, the derivative is $ 6x $.
   - For $ -8x $, the derivative is $ -8 $.
   - So, $ f''(x) = -36x^2 + 6x - 8 $.

5. **Evaluate the second derivative at $ x = 11 $**:
   - Substitute $ x = 11 $ into $ f''(x) $.
   - $ f''(11) = -36(11^2) + 6(11) - 8 $.

6. **Calculate $ 11^2 $**:
   - What is $ 11 \times 11 $?

7. **Continue with the calculation**:
   - If $ 11^2 = 121 $, then $ f''(11) = -36(121) + 66 - 8 $.
   - Can you calculate $ -36(121) + 66 - 8 $?

8. **Determine if it is concave or convex**:
   - If $ f''(11) > 0 $, it is convex.
   - If $ f''(11) < 0 $, it is concave.

What do you think is the sign of $ f''(11) $ after your calculations?