The point on the curve defined by the equation [math: y=(x−4)(x+3)] at which the gradient is [math: −3] has [math: x]-coordinate:

Question

BlackTom AI · Accepted Answer

Differentiate y = (x − 4)(x + 3) to find the gradient function (dy/dx), since the gradient at a point is the derivative evaluated at that x.

1. Relevant knowledge points and concepts
- The gradient (slope) of a curve y = f(x) at a point is given by the derivative dy/dx.
- The product can be expanded or the product rule used; for a quadratic it is simplest to expand first and then differentiate term-by-term.
- To find where the gradient equals a specific value, set the derivative equal to that value and solve for x.

2. Complete reasoning process step by step
- Expand the given function: y = (x − 4)(x + 3).
- Differentiate the expanded form to get dy/dx.
- Set the derivative equal to −3 and solve for x.

3. Calculations with intermediate steps
- Expand: y = x(x + 3) − 4(x + 3) = x^2 + 3x − 4x − 12 = x^2 − x − 12.
- Differentiate: dy/dx = d/dx(x^2) − d/dx(x) − d/dx(12) = 2x − 1 − 0 = 2x − 1.
- Set derivative equal to −3: 2x − 1 = −3.
- Solve: 2x = −3 + 1 = −2, so x = −2/2 = −1.

4. Why the correct answer is correct
- The derivative 2x − 1 gives the slope at any x. Substituting x = −1 yields 2(−1) − 1 = −2 − 1 = −3, which matches the required gradient. Thus x = −1 is the location where the gradient is −3.

5. Why the other options are incorrect
- A: x = 4 → gradient = 2(4) − 1 = 8 − 1 = 7 (not −3).
- B: x = 3 → gradient = 2(3) − 1 = 6 − 1 = 5 (not −3).
- D: x = −3 → gradient = 2(−3) − 1 = −6 − 1 = −7 (not −3).

6. Conclusion
- The x-coordinate where the gradient is −3 is x = −1. This confirms the provided correct answer: C. −1.

The point on the curve defined by the equation [math: y=(x−4)(x+3)] at which the gradient is [math: −3] has [math: x]-coordinate:单项选择题

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