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Find the points on the graph of the function f(x) = x² - 7 that are closest to the point (0, -6).

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Final answer:

To find the points on the graph of the function f(x) = x² - 7 that are closest to the point (0, -6), we can use the concept of minimum distance. The distance between two points in a coordinate plane is given by the distance formula: d = sqrt((x2 - x1)² + (y2 - y1)²). In this case, we want to find the points on the graph that minimize the distance between (x, f(x)) and (0, -6). To do this, we can substitute f(x) = x² - 7 into the distance formula, simplify the expression, and find the x-values that minimize the distance.

Step-by-step explanation:

To find the points on the graph of the function f(x) = x² - 7 that are closest to the point (0, -6), we can use the concept of minimum distance. The distance between two points in a coordinate plane is given by the distance formula: d = sqrt((x2 - x1)² + (y2 - y1)²). In this case, we want to find the points on the graph that minimize the distance between (x, f(x)) and (0, -6). We can substitute f(x) = x² - 7 into the distance formula and simplify the expression.

Step 1: Write the distance formula: d = sqrt((x - 0)² + (f(x) - (-6))²)

Step 2: Substitute f(x) = x² - 7: d = sqrt(x² + (x² - 7 + 6)²)

Step 3: Simplify the expression: d = sqrt(x² + (x² - 1)²)

This expression gives us the distance between any point (x, f(x)) on the graph and the point (0, -6). To find the points on the graph that minimize this distance, we can take the derivative of d with respect to x, set it equal to zero, and solve for x. These values of x will correspond to the x-coordinates of the points on the graph that are closest to the given point.

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