The graphs of g(x) = x³- ax² + 6 and h(x) = 2x² + bx + 3 touch when x = 1. Therefore, the tangent to the curve of g at x = 1 is also the tangent to the curve of h at x = 1.
Determine the coordinates of this point of contact of the two graphs.​

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Step-by-step explanation:

Differentiate each function to find an expression for its gradient (slope of the tangent line) with respect to . Make use of the power rule to find the following:

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The question states that the graphs of and touch at , such that . Therefore:

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On the other hand, since the graph of and coincide at , (otherwise, the two graphs would not even touch at that point.) Therefore:

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Solve this system of two equations for and :

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Therefore, whereas .

Substitute these two values back into the expression for and :

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Evaluate either expression at to find the -coordinate of the intersection. For example, . Similarly, .

Therefore, the intersection of these two graphs would be at .