Multiple Choice
Find the derivative of the given function.
244
views
4
rank
6. Derivatives of Inverse, Exponential, & Logarithmic Functions
Derivatives of Exponential & Logarithmic Functions
Multiple Choice
Identify the local minimum and maximum values of the given function, if any.
,
A
Local maximum of at , Local minimum of at
B
Local maximum of at , No local minima
C
Local minimum of at , No local maxima
D
No local extrema
Verified step by step guidance1
To find the local extrema of the function \( f(t) = t^2 \ln t \), we first need to find its derivative. Use the product rule for differentiation, which states that if \( u(t) = t^2 \) and \( v(t) = \ln t \), then \( (uv)' = u'v + uv' \).
Calculate the derivatives: \( u'(t) = 2t \) and \( v'(t) = \frac{1}{t} \). Substitute these into the product rule to find \( f'(t) = 2t \ln t + t^2 \cdot \frac{1}{t} = 2t \ln t + t \).
Set the derivative \( f'(t) = 2t \ln t + t \) equal to zero to find critical points: \( 2t \ln t + t = 0 \). Factor out \( t \) to get \( t(2 \ln t + 1) = 0 \). Since \( t > 0 \), we solve \( 2 \ln t + 1 = 0 \).
Solve \( 2 \ln t + 1 = 0 \) to find \( \ln t = -\frac{1}{2} \). Exponentiate both sides to solve for \( t \): \( t = e^{-\frac{1}{2}} = \frac{1}{\sqrt{e}} \). This is the critical point.
To determine if this critical point is a local minimum or maximum, use the second derivative test. Find \( f''(t) \) and evaluate it at \( t = \frac{1}{\sqrt{e}} \). If \( f''(t) > 0 \), it's a local minimum; if \( f''(t) < 0 \), it's a local maximum. Calculate \( f''(t) \) and substitute \( t = \frac{1}{\sqrt{e}} \) to conclude.
Related Videos
Related Practice
