Question

If ƒ(x) = x · e^{x(1 – x)}, then ƒ(x) is –

• A. Increasing on$\left\lbrack - \frac{1}{2},1 \right\rbrack$
• B. Decreasing on R
• C. Increasing on R
• D. Decreasing on $\left\lbrack - \frac{1}{2},1 \right\rbrack$

Answer 1

Answer: (A)

Increasing on$\left\lbrack - \frac{1}{2},1 \right\rbrack$

Answer 2

Here ƒ′(x) = x · e^{x(1 – x)} · (1 – 2x) + 1 · e^{x(1 – x)}

ƒ′(x) = e^{x(1 – x)} [x – 2x² + 1]

ƒ′(x) = –e^{x(1 – x)} (x – 1) (2x + 1)

Using number line rule for ƒ′(x) we get, ­ƒ′(x) ≥ 0, when x ∈ $\left\lbrack - \frac{1}{2},1 \right\rbrack$ as shown in figure.

Therefore, (1) is correct option.