The given expression is:

$f(x) = 5x^6 - 18x^5 + 15x^4 - 10$

To find the maximum and minimum values of $f(x)$, follow these steps:

Step 1: Compute the first derivative

The first derivative is:

$f'(x) = 30x^5 - 90x^4 + 60x^3$

Step 2: Factorize $f'(x)$

Factorize the derivative:

$f'(x) = 30x^3(x^2 - 3x + 2)$

Factorize $x^2 - 3x + 2$:

$f'(x) = 30x^3(x - 1)(x - 2)$

Step 3: Critical points

Set $f'(x) = 0$:

$30x^3(x - 1)(x - 2) = 0$

The critical points are:

$x = 0, \, x = 1, \, x = 2$

Step 4: Second derivative test

The second derivative is:

$f''(x) = 150x^4 - 360x^3 + 180x^2$

Evaluate $f''(x)$ at each critical point:

1. At $x = 0$: $f''(0) = 150(0)^4 - 360(0)^3 + 180(0)^2 = 0$ The test is inconclusive for $x = 0$.

2. At $x = 1$: $f''(1) = 150(1)^4 - 360(1)^3 + 180(1)^2 = 150 - 360 + 180 = -30$ Since $f''(1) < 0$, $f(x)$ has a local maximum at $x = 1$.

3. At $x = 2$: $f''(2) = 150(2)^4 - 360(2)^3 + 180(2)^2 = 150(16) - 360(8) + 180(4)$ Simplify: $f''(2) = 2400 - 2880 + 720 = 240$ Since $f''(2) > 0$, $f(x)$ has a local minimum at $x = 2$.

Step 5: Evaluate $f(x)$ at critical points

Compute $f(x)$ at $x = 0, x = 1,$ and $x = 2$:

1. At $x = 0$: $f(0) = 5(0)^6 - 18(0)^5 + 15(0)^4 - 10 = -10$

2. At $x = 1$: $f(1) = 5(1)^6 - 18(1)^5 + 15(1)^4 - 10 = 5 - 18 + 15 - 10 = -8$

3. At $x = 2$: $f(2) = 5(2)^6 - 18(2)^5 + 15(2)^4 - 10$ Simplify: $f(2) = 5(64) - 18(32) + 15(16) - 10 = 320 - 576 + 240 - 10 = -26$

Results

• Local maximum: $f(1) = -8$ at $x = 1$.
• Local minimum: $f(2) = -26$ at $x = 2$.
• At $x = 0$, $f(0) = -10$ (neither max nor min).

Would you like detailed graphs or further analysis?

Related Questions:

1. How can the global maximum and minimum of a polynomial be confirmed?
2. What is the significance of the second derivative test in optimization?
3. How does factoring simplify finding critical points?
4. Could the endpoints of a domain affect the max/min values?
5. How is $f'(x) = 0$ connected to real-world applications?

Tip: Always check endpoints if the domain is restricted, as they may yield absolute extrema.