Let $h (x) = x^{4} - 4 x^{3} + 6 x^{2} - 4 x$ be defined on the closed interval $[0, 2]$ . (i) Verify that $h$ satisfies the hypotheses of Rolle’s theorem on $[0, 2]$ . (ii) Find all numbers $c$ in $(0, 2)$ such that $h^{'} (c) = 0$ .

medium 3 marks Class 12 · Mathematics · Continuity and Differentiability

Answer

The function satisfies Rolle’s theorem and

c = 1

is the only point in

(0, 2)

with

h^{'} (c) = 0

The correct answer is The function satisfies Rolle’s theorem and $c = 1$ is the only point in $(0, 2)$ with $h^{'} (c) = 0$ . — The solution $c = 1$ lies in the open interval $(0, 2)$ , satisfying Rolle’s conclusion.

Step-by-step solution

Check continuity on $[0, 2]$ : $h (x)$ is a polynomial, hence continuous everywhere, in particular on $[0, 2]$ .
Check differentiability on $(0, 2)$ : polynomials are differentiable everywhere, so $h$ is differentiable on $(0, 2)$ .
Evaluate the endpoint values: $h (0) = 0$ and $h (2) = 2^{4} - 4 \cdot 2^{3} + 6 \cdot 2^{2} - 4 \cdot 2 = 16 - 32 + 24 - 8 = 0$ .
Since $h (0) = h (2)$ , all three hypotheses of Rolle’s theorem are met.
Differentiate: $h^{'} (x) = 4 x^{3} - 12 x^{2} + 12 x - 4$ .
Factor the derivative: $h^{'} (x) = 4 (x^{3} - 3 x^{2} + 3 x - 1) = 4 (x - 1)^{3}$ .
Set $h^{'} (c) = 0$ : $4 (c - 1)^{3} = 0 \Rightarrow (c - 1)^{3} = 0 \Rightarrow c = 1$ .
The solution $c = 1$ lies in the open interval $(0, 2)$ , satisfying Rolle’s conclusion.

Common mistake: Incorrect factoring of

h^{'} (x)

and missing the triple root at

x = 1

Concept tested: Rolle’s theorem and finding critical points

Published 7 September 2026 · How this solution was prepared and checked

← All Continuity and Differentiability questions

Let h(x)=x4−4x3+6x2−4x be defined on the closed interval [0,2]. (i) Verify that h satisfies the hypotheses of Rolle’s theorem on [0,2]. (ii) Find all numbers c in (0,2) such that h′(c)=0.

Step-by-step solution

Let $h (x) = x^{4} - 4 x^{3} + 6 x^{2} - 4 x$ be defined on the closed interval $[0, 2]$ . (i) Verify that $h$ satisfies the hypotheses of Rolle’s theorem on $[0, 2]$ . (ii) Find all numbers $c$ in $(0, 2)$ such that $h^{'} (c) = 0$ .