the function f(x)= -2x³ + 36x² - 192x + 3 has...
the function f(x)= -2x³ + 36x² - 192x + 3 has one local minimum and one local maximum. this function has a local minimum at x = with value and a local maximum at x = with value question help: video written example
Answer
# Explanation:
## Step1: Find the derivative
Differentiate $f(x)=-2x^{3}+36x^{2}-192x + 3$ using power - rule. $f'(x)=-6x^{2}+72x - 192$.
## Step2: Set derivative equal to 0
Solve $-6x^{2}+72x - 192 = 0$. Divide by -6: $x^{2}-12x + 32=0$. Factor: $(x - 4)(x - 8)=0$. So $x = 4$ and $x = 8$.
## Step3: Use second - derivative test
Differentiate $f'(x)$ to get $f''(x)=-12x + 72$.
When $x = 4$, $f''(4)=-12\times4 + 72=24>0$, so $f(x)$ has a local minimum at $x = 4$.
When $x = 8$, $f''(8)=-12\times8+72=-24<0$, so $f(x)$ has a local maximum at $x = 8$.
## Step4: Find function values
For local minimum at $x = 4$, $f(4)=-2\times4^{3}+36\times4^{2}-192\times4 + 3=-2\times64+36\times16-768 + 3=-128+576-768 + 3=-317$.
For local maximum at $x = 8$, $f(8)=-2\times8^{3}+36\times8^{2}-192\times8 + 3=-2\times512+36\times64-1536 + 3=-1024+2304-1536 + 3=-253$.
# Answer:
Local minimum at $x = 4$ with value $-317$.
Local maximum at $x = 8$ with value $-253$.