Exams › GATE › Engineering Mathematics

Consider a differentiable function \(f(x)\) on the set of real numbers such that \(f(-1)=0\) and \(|f'(x)|\le 2\). Which one of the following inequalities is necessarily true for all \(x\in[-2,2]\)?

1. \(f(x)\le \frac{1}{2}|x+1|\)
2. \(f(x)c2|x+1|\)
3. \(f(x)c2\frac{1}{2}|x|\)
4. \(f(x)c2|x|\)

Correct answer: \(f(x)c2|x+1|\)

Solution

By the Mean Value Theorem, \(|f(x)-f(-1)|\le 2|x-(-1)|=2|x+1|\). Since \(f(-1)=0\), this becomes \(|f(x)|\le 2|x+1|\). Among the given options, the intended correct inequality is \(f(x)\le 2|x+1|\).

Related GATE Engineering Mathematics questions

• The value of \(\lim_{x\to\infty}\left(x-\sqrt{x^2+x}\right)\) is equal to
• Consider the polynomial f(x) = x^3 - 6x^2 + 11x - 6 on the domain S: 1 \le x \le 3. Let f'(x) and f''(x) be the first and second derivatives. Consider the following statements: I. The given polynomial is zero at the boundary points x = 1 and x = 3. II. There exists one local maximum of f(x) within the domain S. III. The second derivative f''(x) > 0 throughout the domain S. IV. There exists one local minimum of f(x) within the domain S. The correct option is:
• The optimum value of the function $f(x)=x^2-4x+2$ is
• What is the value of $\lim_{(x,y)\to(0,0)} \dfrac{xy}{x^2+y^2}$?
• The angle of intersection of the curves \(x^2=4y\) and \(y^2=4x\) at the point \((0,0)\) is
• The quadratic approximation of \(f(x)=x^3-3x^2-5\) at \(x=0\) is

⚔️ Practice GATE Engineering Mathematics free + battle 1v1 →