Critical points Find the critical points of the function ƒ(x) = sinh² x cosh x.

Verified step by step guidance

1. Recall that critical points occur where the derivative of the function is zero or undefined. So, first, find the derivative of the function \(f(x) = \sinh^{2} x \cdot \cosh x\).
2. Use the product rule for differentiation: if \(f(x) = u(x) v(x)\), then \(f'(x) = u'(x) v(x) + u(x) v'(x)\). Here, let \(u(x) = \sinh^{2} x\) and \(v(x) = \cosh x\).
3. Find the derivatives of \(u(x)\) and \(v(x)\) separately: \(u'(x) = 2 \sinh x \cdot \cosh x\) (using the chain rule) and \(v'(x) = \sinh x\).
4. Substitute these into the product rule formula to get \(f'(x) = (2 \sinh x \cosh x)(\cosh x) + (\sinh^{2} x)(\sinh x)\), then simplify the expression.
5. Set the derivative \(f'(x)\) equal to zero and solve for \(x\) to find the critical points. Also, check where \(f'(x)\) might be undefined, although hyperbolic functions are defined everywhere.

Key Concepts

Hyperbolic Functions

Hyperbolic functions, such as sinh(x) and cosh(x), are analogs of trigonometric functions but based on hyperbolas. They have unique properties and derivatives: the derivative of sinh(x) is cosh(x), and the derivative of cosh(x) is sinh(x). Understanding these functions is essential for differentiating and analyzing the given function.

Critical Points

Critical points of a function occur where its derivative is zero or undefined. These points are important because they can indicate local maxima, minima, or points of inflection. Finding critical points involves computing the derivative and solving for values of x that satisfy these conditions.

Product and Chain Rule in Differentiation

The given function is a product of sinh²(x) and cosh(x), so applying the product rule is necessary to differentiate it. Additionally, since sinh²(x) is a composite function, the chain rule is used to differentiate it correctly. Mastery of these differentiation rules is crucial to find the derivative accurately.