---
title: "Verify Rolle's theorem: x(x+3)/e^(x/2)"
subject: "Mathematics I"
chapter: 1
key: "semester_1/mathematics_i:rolle-verify-x-x-plus-3"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/rolle-verify-x-x-plus-3
site: BCT Engineering
---

# Verify Rolle's theorem: x(x+3)/e^(x/2)

Mathematics I · Chapter 1: Derivatives and their Applications

- Asked in: 2072 Kartik; 2070 Asar (asked 2 times)
- Video: [Verifying Rolle's theorem of function f(x)=x(x+3)e^-x/2 in (-3,0) Problem in Mean Value Theorem.](https://www.youtube.com/watch?v=rLJnSps39qs) — Arudra Maths Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/rolle-verify-x-x-plus-3
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-1

## Question

State Rolle's Theorem and verify the theorem for f(x) = x(x+3)/e^(x/2); x ∈ [−3, 0].

## Short answer

Rolle's theorem: if f is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then ∃c∈(a,b) with f'(c)=0. For f(x)=x(x+3)e^{-x/2} on [-3,0], f(-3)=f(0)=0, and f'(c)=0 gives c=−2 in (-3,0), verifying the theorem.

## Answer

#### Rolle's theorem
If f is continuous on [a, b], differentiable on (a, b) and f(a) = f(b), then there is at least one c in (a, b) such that f′(c) = 0.

#### Verification
f(x) = x(x + 3) e^(−x/2) on [−3, 0].

1. **Continuity and differentiability:** f is a product of a polynomial and an exponential, so it is continuous on [−3, 0] and differentiable on (−3, 0).
2. **Equal end values:** f(−3) = (−3)(0)e^(3/2) = 0 and f(0) = 0·3·1 = 0. So f(−3) = f(0).
3. **Derivative:**

```text
f′(x) = e^(−x/2) [ (2x + 3) − x(x+3)/2 ]
      = e^(−x/2) (−x² + x + 6)/2
      = −(1/2)(x − 3)(x + 2) e^(−x/2)
```
Setting f′(c) = 0 gives c = 3 or c = −2. Only c = −2 lies in (−3, 0).

**Hence Rolle's theorem is verified with c = −2.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which set of conditions must be checked in Rolle's theorem?
   A. Only f(a) = f(b)
   B. f' > 0 and f(a) = f(b)
   C. Continuity on [a,b], differentiability on (a,b), f(a) = f(b)
   D. Continuity on (a,b) and f(a) = 0
   Answer: C. These three conditions guarantee f'(c) = 0 for some c. (Key point: Rolle's conditions)

2. For f(x) = x(x+3)e^(-x/2) on [-3, 0], f(-3) and f(0) are:
   A. both 0
   B. 0 and 3
   C. e^(3/2) and 0
   D. both 3
   Answer: A. Both factors x and (x+3) kill the value at the ends. (Key point: End values)

3. f'(x) = -(1/2)(x-3)(x+2)e^(-x/2) vanishes at:
   A. x = -3 and 0
   B. x = 2 and -3
   C. x = 3 and x = -2
   D. x = 0 only
   Answer: C. Roots are 3 and -2. (Key point: Derivative root)

4. Which root is c?
   A. c = 0
   B. Both
   C. c = 3
   D. c = -2 because it lies in (-3, 0)
   Answer: D. Only -2 lies in the open interval (-3, 0). (Key point: Choosing c)

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