A) Which statement is true regarding the relationship between critical points and extrema? (i) If...

Question:

A) Which statement is true regarding the relationship between critical points and extrema?

(i) If {eq}f(a) {/eq} is a minimum or maximum, then {eq}x = a {/eq} must be a critical point.

(ii) If {eq}x = a {/eq} is a critical point, then {eq}f(a) {/eq} must be a minimum or maximum.

(iii) Not every minimum or maximum value is a critical point.

B) Write the derivative of {eq}\sinh(2x) {/eq} as exponentials.

Critical Points

Critical points of a function are the points where the function is either 'not differentiable' or its derivative at those points is zero. For a differentiable curve, relative maximum or relative minimum can only exist at critical points.

Let {eq}y=f(x) {/eq} be any differentiable curve. The point {eq}x=x_0 {/eq} is its critical point iff {eq}f'(x_0)=0 {/eq}.

Answer and Explanation:

A) Correct Option: (iii)

Consider the function {eq}f:S \to \mathbb{R} {/eq} defined by {eq}f(x) = x^2 {/eq} where {eq}S = (0,3] {/eq}. Its maximum occurs at {eq}x = 3 {/eq} but {eq}x= 3 {/eq} is not a critical point as {eq}f'(3) = 2*3 = 6\neq 0 {/eq}.


B) By definition {eq}\sinh 2x = \frac{e^{2x}-e^{-2x}}{2} \,\,\Rightarrow\,\,(\sinh 2x)' = \frac{d(\sinh 2x)}{dx} =\frac{1}{2}(\frac{d(e^{2x})}{dx} - \frac{d(e^{-2x})}{dx}) = e^{2x} + e^{-2x} {/eq}


Learn more about this topic:

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Finding Critical Points in Calculus: Function & Graph

from CAHSEE Math Exam: Tutoring Solution

Chapter 8 / Lesson 9
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