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Chapter 2: Problem 40

Find the minimum value of $$ f(x)=8^{\sin ^{-1} x}+8^{\cos ^{-1} x} $$

Short Answer

Expert verified
The minimum value of function \(f(x)=8^{\sin^{-1}x}+8^{\cos^{-1}x}\) is approximately 32.

Step by step solution

01

Identify key properties

One important property to remember here is \(\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}\). This is deduced from the fact that sine and cosine are complementary. So, \(8^{\sin ^{-1} x}+8^{\cos ^{-1} x} = 8^{\frac{\pi}{2}} + 8^{\frac{\pi}{2}}\).
02

Simplify the expression

After substituting the right hand side of the equation by \(\frac{\pi}{2}\), the function \(f(x)\) simplifies to \(2 \cdot 8^{\frac{\pi}{2}}\), which is the sum of two identical terms.
03

Compute the minimum value

The given function is increasing with its minimum occurring at its smallest value of x. Given \(f(x) = 2 \cdot 8^{\frac{\pi}{2}}\), calculating the expression gives a value of approximately 31.92 or rounded 32, which is the minimum value of the function.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inverse Trigonometric Functions
Inverse trigonometric functions allow us to work backwards from regular trigonometric outputs to angles. They are crucial when angles must be determined from known sine or cosine values. For example, \( \sin^{-1} x \) is the angle whose sine is \( x \). The key property used in this problem is \( \sin^{-1}x + \cos^{-1}x = \frac{\pi}{2} \). This stems from how sine and cosine values represent the same angle on the unit circle in complementary forms. It's essential to understand this relation as it simplifies many trigonometric expressions.
  • Key Property: Sine and cosine are related such that their inverse functions sum up to \( \frac{\pi}{2} \).
  • Understanding the Circle: Picture a right triangle on the unit circle, where sine and cosine define two sides of the triangle; their roles swap depending on the complementary nature of the angles.
Properties of Functions
Understanding the properties of the function \( f(x) = 8^{\sin^{-1} x} + 8^{\cos^{-1} x} \) involves exploring how such functions behave. Function properties like monotonicity (always increasing or decreasing) play a key role in finding extremes like minima or maxima. In this scenario, the function is simplified using its inherent trigonometric property, which turns it into constants multiplied by powers of 8.
  • Simplification Method: By converting trigonometric terms using known identities, the function can be reduced to a standard algebraic form.
  • Monotonic Behavior: The function's nature of increasing helps establish where the minimum occurs, effectively at the lower boundary of the domain.
Minimization Problems
Minimization problems involve finding the smallest value of a function within a given domain. For \( f(x) \), recognizing it is increasing allows simplification to find the minimum without requiring calculus-based techniques. Here, since the function is expressed as a sum of powers, simplification leads directly to a constant expression with minimal effort. There are vital aspects common to minimization problems:
  • Recognizing Patterns: Identify symmetrical or complementary features, as seen with trigonometric identities, to aid in simplification.
  • Simplification: Using known identities reduces the complexity of calculations.
  • Computational Checks: Once simplified, calculate the expression to estimate quantitatively, ensuring it makes sense within the problem's context.
With these insights, the calculation \( 2 \cdot 8^{\frac{\pi}{2}} \) gives a value of approximately 31.92, typically rounded to 32, indicating how the output directly reflects the problem-solving process.

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