Exponential and trigonometric functions practice question 12
What is the largest open interval on which is increasing?
Increasing is a question about the derivative, not about where itself is positive. Where is positive?
Product rule: . Factor out .
is always positive, so the sign of is the sign of the other factor.
Theory UBC Math 100 §3.5CLP-1 §2.7Key results
Worked solution
Strategy
Differentiate, factor out the exponential (which never changes sign), and read the sign of the remaining factor.
Working
Product rule, then factor.
ex > 0 for every x, so only 1 + x decides the sign.
Answer
The answer is x greater than negative 1.
Check
f(−2) ≈ −0.271, f(−1) ≈ −0.368 and f(0) = 0, so f dips to its lowest point at x = −1 and climbs after it.
Where this goes wrong
Answering (0, ∞) because f(x) is positive there. Where a function is positive and where it is increasing are different questions; the second is about the derivative.
The idea behind it
The exponential and trigonometric functions lesson, a figure to move, in a few steps
Leans on
The product rule, from Differentiation rules
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