Let f(x)=x^3+1 a. Find the points on the graph of f where...

Let $f(x)=x^{3}+1$
a. Find the points on the graph of $f$ where the slope of the tangent line is equal to 12 .
b. Find the equation(s) of the tangent line(s) of part (a).
c. Sketch the graph of $f$ showing the tangent line(s).

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Key Concepts

Cubic Functions

Cubic functions are polynomial functions of degree three that can display a range of behaviors including multiple turning points and inflection points. Understanding their nature helps in analyzing their graphs, detecting symmetry, and anticipating the number and types of tangent lines that can intersect the curve under different conditions.

Point-Slope Form

The point-slope form of a line is a way to express the equation of a line when you know a specific point on the line and its slope. This form, given by y - y? = m(x - x?), is particularly useful in problems involving tangent lines, where these parameters are extracted from the function and its derivative.

Derivative

The derivative is a core concept in calculus that measures the rate at which a function changes at any point. It provides the slope of the tangent line to the function's graph at a given point and is foundational for understanding optimization, motion, and approximation in various contexts.

Tangent Line

A tangent line is a straight line that touches a curve at a single point and has the same slope as the curve at that point. It is used to approximate the behavior of the function near that point and is essential in linear approximations and differential calculus.

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