A function has a second‐order derivatives 1) If its graph passes through the point and at that point the tangent to the grraph , then the function is

The correct answer is:

The given second order derivative of function is f"(x) = 6(x - 1). To find the first derivative, we will integrate the second derivative.

We will integrate the second derivative.

We know that the first derivative of a function at certain point gives us tangent at that point.
The equation of tangent at the point of derivative is (y - y₁) = f'(x)(x - x1)
We will find the value of derivative at the point (2,1). We will substitute x = 2 in the first order derivative.

We will substitute the values of f'(x) and the point in the equation of the tangent.
(y - 1) = C(x - 2)
y - 1 = Cx - 2C
y = Cx - 2C + 1
y = Cx - (2C - 1)
Comparing with the given equation of tangent of the curve
y = 3x - 5
Cx = 3x
So, C = 3
We will substitute the value of C in the equation of first order derivative.

Now, we will integrate the first order derivative to get the value of function.

To find the value of the constant, we will substitute the value of the point (2,1) in the function.

y = x³ - 3x² + 3x - 1
The given function is the expansion of formula (x - 1)³
y = (x - 1)³
This is the required function.

For such questions, the important part is integration. We should know the methods to integrate the derivate. We should know the properties of a tangent.