Q.32 The maximum value of the function f(x) = 3x2 − 2x3 for x > 0 is ________.
Step-by-Step Solution
The given function is:
f(x) = 3x² − 2x³
Compute the first derivative:
f′(x) = 6x − 6x² = 6x(1 − x)
Set f′(x) = 0 to find critical points:
- x = 0
- x = 1
Since x > 0, consider x = 1.
Second derivative:
f′′(x) = 6 − 12x
At x = 1:
f′′(1) = −6 < 0
This confirms a local maximum.
Evaluate at x = 1:
f(1) = 3(1)² − 2(1)³ = 1
As x → 0⁺, f(x) → 0 and as x → ∞, f(x) → −∞.
Therefore, the global maximum for x > 0 is 1.
Options Insight
Common mistakes include:
- Using x = 0 (excluded since x > 0).
- Ignoring the second derivative test.
- Missing domain constraints.
Graph Behavior
The curve rises to 1 at x = 1 and then decreases toward negative infinity.
