Question

The minimum value of the slope of the tangent to curve $y = x^3 - 3x^2 + 2x + 93$ is

• A. 1
• B. -1
• C. 2
• D. -2

Hint:

For a quadratic $ax^2 + bx + c$, the minimum value is reached at $x = -b/2a$ and the value is $-\Delta/4a$.

Answer 1

The Correct Option is B

Step 1: Find the Slope Function
Slope $m = \frac{dy}{dx} = 3x^2 - 6x + 2$.

Step 2: Minimize the Slope
To find the minimum of $m$, find $\frac{dm}{dx}$ and set it to zero:
$\frac{dm}{dx} = 6x - 6 = 0 \implies x = 1$.

Step 3: Verification
$\frac{d^2m}{dx^2} = 6$ (positive), so $x=1$ is a point of minima for the slope.

Step 4: Calculate Value
At $x = 1$, $m = 3(1)^2 - 6(1) + 2 = 3 - 6 + 2 = -1$.
Final Answer: (B)