Question

an equation for the line tangent to the graph of the differentiable function f at x = 2 is y = 9x - 12. which of the following statements must be true? i. f(0)= - 12 ii. f(2)=6 iii. f(2)=9 a none b i and ii only c ii and iii only d i, ii, and iii

Explanation:

Step1: Recall tangent - line properties

The equation of the tangent line to the graph of $y = f(x)$ at $x = a$ is given by $y - f(a)=f^{\prime}(a)(x - a)$ or $y=f^{\prime}(a)x+(f(a)-af^{\prime}(a))$. The slope of the tangent line to $y = f(x)$ at $x = a$ is $f^{\prime}(a)$, and the point $(a,f(a))$ lies on both the function $y = f(x)$ and its tangent - line.

Step2: Analyze statement I

The equation of the tangent line is $y = 9x-12$. There is no information from the tangent - line equation $y = 9x - 12$ at $x = 2$ that can be used to find $f(0)$. The tangent line is only relevant at the point of tangency $x = 2$. So, $f(0)
eq - 12$ in general.

Step3: Analyze statement II

Since the tangent line to $y = f(x)$ at $x = 2$ is $y = 9x-12$, when $x = 2$, we substitute $x = 2$ into the tangent - line equation. $y=9\times2 - 12=18 - 12 = 6$. The point $(2,f(2))$ lies on the tangent line, so $f(2)=6$.

Step4: Analyze statement III

The slope of the tangent line $y = 9x-12$ is $9$. The slope of the tangent line to the function $y = f(x)$ at $x = 2$ is $f^{\prime}(2)$. Since the slope of the line $y = 9x-12$ is $9$, we have $f^{\prime}(2)=9$.

Answer:

C. II and III only