Question

1. let f be the function defined by f(x) = 4x³ - 5x + 3. which of the following is an equation of the line tangent to the graph of f at the point where x = -1?

Explanation:

Step1: Find the derivative of the function

The derivative of \(f(x)=4x^{3}-5x + 3\) is \(f^{\prime}(x)=12x^{2}-5\) (using the power rule \((x^{n})^\prime=nx^{n - 1}\)).

Step2: Find the slope of the tangent line at \(x=-1\)

Substitute \(x = - 1\) into \(f^{\prime}(x)\): \(f^{\prime}(-1)=12(-1)^{2}-5=12 - 5=7\).

Step3: Find the \(y\) - coordinate of the point on the function at \(x=-1\)

Substitute \(x=-1\) into \(f(x)\): \(f(-1)=4(-1)^{3}-5(-1)+3=-4 + 5+3=4\).

Step4: Use the point - slope form \(y - y_{1}=m(x - x_{1})\)

Here \(m = 7\), \(x_{1}=-1\), \(y_{1}=4\). The equation is \(y - 4=7(x + 1)\), which simplifies to \(y=7x+11\).

Answer:

\(y = 7x + 11\)