The function f(x)=-x^4+6x^2+8xhasx=-1 as a critical number. The point (-1,-1) on the graph of y=f(x) is (i) local maximum (ii) local minimum (iii) point of inflexion (i) is true (ii) is true. (iii) is true. (i), (ii), (iii) are all true. none of these is true.

## Step 1: Find the first derivative<br />### We need to find the first derivative of the function $f(x) = -x^4 + 6x^2 + 8x$. Using the power rule, we get $f'(x) = -4x^3 + 12x + 8$.<br /><br />## Step 2: Find the second derivative<br />### Now, we find the second derivative of the function, which is $f''(x) = -12x^2 + 12$.<br /><br />## Step 3: Evaluate the second derivative at the critical point<br />### We are given that $x = -1$ is a critical number. We evaluate the second derivative at this point: $f''(-1) = -12(-1)^2 + 12 = -12 + 12 = 0$.<br /><br />## Step 4: Analyze the second derivative<br />### Since the second derivative at $x = -1$ is 0, the second derivative test is inconclusive. We need to analyze the behavior of the function around the critical point.<br /><br />## Step 5: Further analysis using the first derivative<br />### Let's analyze the sign of the first derivative around $x = -1$. For $x = -2$, $f'(-2) = -4(-8) + 12(-2) + 8 = 32 - 24 + 8 = 16 > 0$. For $x = 0$, $f'(0) = 8 > 0$. For $x = -0.5$, $f'(-0.5) = -4(-0.125) + 12(-0.5) + 8 = 0.5 - 6 + 8 = 2.5 > 0$. Since the first derivative doesn't change sign around $x=-1$, and $f'(-1) = -4(-1) + 12(-1) + 8 = 4 - 12 + 8 = 0$, we have a possible inflection point.<br /><br />## Step 6: Check for inflection point<br />### Since $f''(-1) = 0$, we check the concavity around $x = -1$. We found $f''(x) = -12x^2 + 12$. For $x < -1$ (e.g., $x = -2$), $f''(-2) = -12(4) + 12 = -36 < 0$ (concave down). For $x > -1$ (e.g., $x = 0$), $f''(0) = 12 > 0$ (concave up). Since the concavity changes around $x = -1$, it is an inflection point.