The triangle formed by the tangent to the parabola y=x^2 at...

The triangle formed by the tangent to the parabola $y=x^{2}$ at the point $x=x_{0}\left(x_{0} \in[1,2]\right)$ the y-axis and the straight line $\mathrm{y}=\mathrm{x}_{0}{ }^{2}$ has the greatest area if $\mathrm{x}_{0}=$
(a) 1
(b) $\frac{4}{3}$
(c) $\frac{3}{2}$
(d) 2

Key Concepts

Derivative

The derivative of a function measures its rate of change and is used to determine the slope of the tangent line at any point on the curve. This is fundamental in calculus for understanding how functions behave locally and for constructing the equation of the tangent line.

Equation of Tangent Line

Using the derivative, the tangent line to a curve at a specific point is found by applying the point-slope formula. This linear approximation is crucial in problems where the behavior of a curve near a point is analyzed or where it interacts with other geometric objects.

Area of a Triangle

The calculation of a triangle's area, typically using the formula 1/2 × base × height, is an essential geometric concept. In coordinate geometry, determining the area can involve finding intersections of lines and using coordinate distances, which is useful in optimization problems involving geometric shapes.

Optimization

Optimization in mathematics involves finding the maximum or minimum value of a function. In calculus, this often means taking derivatives to determine critical points where the function might achieve an extreme value, and then analyzing these points to find the optimal solution.

Coordinate Geometry

Coordinate geometry integrates algebra and geometry by representing geometric figures using equations. This allows for the computation of intersections, distances, and areas using algebraic methods, making it a powerful tool in solving complex geometric problems.

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