Find c such that y=4x+2 is tangent to the curve `y=1/3 x^3+c `

The slope of the tangent line to a curve is given by the value of the first derivative.

Since the line is tangent to the curve, we find where the tangent to the curve has value 4...

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Find c such that y=4x+2 is tangent to the curve `y=1/3 x^3+c `

The slope of the tangent line to a curve is given by the value of the first derivative.

Since the line is tangent to the curve, we find where the tangent to the curve has value 4 (the slope of the line):

`y'=x^2 ` and x^2=4 ==> x=2 or -2.

Suppose x=2; then y=8/3+c. The value of the line at x=2 is 10 so we have 8/3+c=10 or c=22/3

Suppose x=-2; then y=-8/3+c. The value of the line at x=-2 is -6 so -8/3x+c=-6 or c=-10/3

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c can be either -10/3 or 22/3

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The graph of y=1/3x^3+22/3 in blue, y=1/3x^3-10/3 in green, and y=4x+2 in red: