Let's tackle the questions one by one.

6. Gunakan uji turunan pertama untuk menentukan ekstrem lokal pada soal-soal berikut dan tentukan jenisnya.

The task here is to use the first derivative test to find and classify the local extrema (either minima or maxima) of the given functions.

• 6(a). $f(x) = x^3 + x^2 - x + 2$

  1. Find the derivative: $f'(x) = 3x^2 + 2x - 1$

  2. Find critical points by setting the derivative equal to zero: $3x^2 + 2x - 1 = 0$ Solve the quadratic equation using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-2 \pm \sqrt{(2)^2 - 4(3)(-1)}}{2(3)} = \frac{-2 \pm \sqrt{4 + 12}}{6} = \frac{-2 \pm \sqrt{16}}{6}$ $x = \frac{-2 \pm 4}{6}$ Hence, the critical points are $x = \frac{1}{3}$ and $x = -1$.

  3. First derivative test: Check the sign of $f'(x)$ around the critical points.

     • For $x = \frac{1}{3}$, check intervals like $(-\infty, -1)$, $(-1, \frac{1}{3})$, and $(\frac{1}{3}, \infty)$.

     Based on the sign changes of the derivative, we can classify whether the critical points are minima or maxima.

• 6(b). $f(x) = \frac{-2x}{(1+x^2)}$

  1. Find the derivative: Use the quotient rule or simplify: $f'(x) = \text{use quotient or chain rule here}$

  2. Critical points: Solve $f'(x) = 0$ to find critical points.

  3. Classify extrema using the first derivative test.

• 6(c). $f(x) = x^3 - 2x^2 + x + 1$

  1. Find the derivative: $f'(x) = 3x^2 - 4x + 1$

  2. Critical points: Solve the quadratic equation $3x^2 - 4x + 1 = 0$.

  3. Classify extrema using the first derivative test.

7. Gunakan uji turunan kedua untuk menentukan ekstrem lokal pada soal-soal nomor 6 dan tentukan jenisnya.

If the second derivative exists, we can apply the second derivative test to classify the critical points as minima, maxima, or points of inflection.

• For each function from question 6:
  1. Find the second derivative $f''(x)$.
  2. Evaluate $f''(x)$ at the critical points found from the first derivative.
  3. If $f''(x) > 0$ at a critical point, it is a local minimum; if $f''(x) < 0$, it is a local maximum.

8. Sketsakan kurva fungsi berikut ini

You are asked to sketch the curves of the following functions:

• 8(a). $f(x) = |x|(x^2 - 1)$

  • This is an absolute value function combined with a cubic polynomial, so you need to sketch how the absolute value affects the shape of the graph.

• 8(b). $f(x) = x^4 - 2x^2 + 1$

  • This is a quartic function. You can determine the overall shape and behavior by finding critical points and inflection points.

• 8(c). $f(x) = |x|(x+1)^2$

  • Similar to (a), this involves an absolute value combined with a quadratic function. Sketch how the absolute value affects the symmetry and shape of the graph.

Would you like me to proceed with detailed calculations for each part or provide further clarifications on specific questions?

Here are 5 follow-up questions:

1. Would you like a step-by-step solution for finding the critical points of each function?
2. Should I proceed with the second derivative test for each function in question 6?
3. Would you like help with the sketching of each function in question 8?
4. Should I explain the quotient and chain rules for part 6(b)?
5. Would you like more details on how to classify critical points using the first derivative test?

Tip:

When sketching curves, always start by analyzing the domain, critical points, behavior at infinity, and symmetries.