Question

The line y = ax + b is a tangent to the curve y = 2x³ − 5x² − 3x + c at the point (2, 6). Find the  values of the constants a, b and c.

Solution

We are given equation of the line as;

We are given equation of the curve as;

It is given that line is tangent to the curve at point P(2,6).

The slope of a curve at a particular point is equal to the slope of the tangent to the curve at the same point;

First, we find slope of the line.

Slope-Intercept form of the equation of the line;

Where is the slope of the line.

Therefore, in the given equation of the line , the slope of the line is;

Next, we find slope of the curve at point P(2,6).

Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve with respect to  is:

We are given equation of the curve;

Therefore;

 

Rule for differentiation of  is:

Rule for differentiation of  is:

Rule for differentiation of  is:

To find slope of the tangent to the curve at point we need gradient of the curve at the same  point.

Gradient (slope) of the curve at the particular point is the derivative of equation of the curve at that  particular point.

Gradient (slope) of the curve at a particular point can be found by substituting x- coordinates of that point in the expression for gradient of the curve;

Therefore, to find the gradient at the curve at the point we substitute  in;

Now we have slope of the curve as;

We can equate slope of the tangent to the curve at this point of the curve i.e., point ;

We are given equation of the line as;

We can substitute the value of ;

Since this line is tangent to the curve at point P(2,6), the point lies on both the curve and the line.

If a point lies on the line or the curve, the coordinates of the point must satisfy the equation of the  line or the curve.

Therefore, we substitute coordinates of the point P(2,6) in equation of the line;

We also substitute coordinates of the point P(2,6) in equation of the curve;