Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima.

f(x) = eˣ(x - 2)²

검증된 단계별 안내

First, find the first derivative of the function f(x) = e^x(x - 2)^2. Use the product rule: if u(x) = e^x and v(x) = (x - 2)^2, then f'(x) = u'(x)v(x) + u(x)v'(x).

Calculate the derivatives: u'(x) = e^x and v'(x) = 2(x - 2). Substitute these into the product rule to find f'(x).

Set the first derivative f'(x) equal to zero to find the critical points. Solve the equation for x to find the values where the slope of the tangent is zero.

Next, find the second derivative f''(x) by differentiating f'(x) again. This will involve using the product rule and chain rule as necessary.

Evaluate the second derivative at each critical point found. Use the Second Derivative Test: if f''(x) > 0, the point is a local minimum; if f''(x) < 0, the point is a local maximum. If f''(x) = 0, the test is inconclusive.

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주요 개념

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Critical Points

Critical points of a function occur where its first derivative is zero or undefined. These points are essential for identifying potential local maxima and minima, as they represent locations where the function's slope changes. To find critical points, one must differentiate the function and solve for the values of x that satisfy the condition f'(x) = 0.

Second Derivative Test

The Second Derivative Test is a method used to classify critical points as local maxima, local minima, or saddle points. It involves evaluating the second derivative of the function at the critical points. If f''(x) > 0, the point is a local minimum; if f''(x) < 0, it is a local maximum; and if f''(x) = 0, the test is inconclusive.

Exponential and Polynomial Functions

The function f(x) = eˣ(x - 2)² combines an exponential function, eˣ, and a polynomial function, (x - 2)². Understanding the behavior of these types of functions is crucial, as exponential functions grow rapidly, while polynomial functions can have varying degrees of growth based on their degree. This knowledge helps in analyzing the overall shape and critical points of the function.

Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima.f(x) = eˣ(x - 2)²