# 11.7.5. Applications on Derivatives

If the curve touches the straight line at the point , find the two values of and .

• A,
• B,
• C,
• D,

### Example

If the curve touches the straight line at the point , find the two values of and .

### Solution

If the curve touches, but does not cross the straight line at the point , then the straight line is a tangent to the curve at this point, and hence the curve and the straight line have the same slope.

The slope of the straight line is 7.

The slope of the curve can be found by differentiating.

The slopes are equal when , so substituting this value leads to

The point is on both the straight line and the curve, and therefore substituting these values for and into the equation of the curve will yield a second equation in and .

Hence, there are two linear equations for and which can be solved simultaneously. Multiplying the second equation by 2 gives

Adding this equation to the first gives

Substituting this value of into the second equation gives

Therefore, the solution is , .

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