Question:
Let \(f(x)=x^2+ax+b\) and \(g(x)=f(x+1)-f(x-1)\). If \(f(x)≥0\) for all real \(x\), and \(g(20)=72\), then the smallest possible value of \(b\) is
Let \(f(x)=x^2+ax+b\) and \(g(x)=f(x+1)-f(x-1)\). If \(f(x)≥0\) for all real \(x\), and \(g(20)=72\), then the smallest possible value of \(b\) is
Updated On: Sep 17, 2024
- 1
- 16
- 0
- 4
Hide Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
The correct answer is (D): \(4\)
\(f(x)=x^2+ax+b\)
\(g(x)=f(x+1)-f(x-1)\)
=\({(x+1)^2+a(x+1)+b}-{(x-1)^2+a(x-1)+b}\)
\(g(x)=4x+2a\)
\(g(20)=72\)
\(80+2a=72 ⇒ a=-4\)
\(∴ f(x)=x^2-4x+b\)
\(f(x)=(x-2)^2+b-4\)
When \(b≥4 f(x)≥0\) for all \(x\)
\(∴\) The minimum value of b is \(4\)
Was this answer helpful?
0
0
Top Questions on Functions
- Suppose \(f(x,y)\) is a real-valued function such that \(f(3x+2y,2x-5y)=19x\), for all real numbers \(x\) and \(y\) . The value of x for which \(f(x,2x) = 27\) , is
- Let r be a real number and \(f(x) = \begin{cases} 2x - r & \text{if } x \geq r \\ r & \text{if } x < r \end{cases}\).Then, the equation \(f(x)=f(f(x))\) holds for all real values of x where
- Suppose for all integers x, there are two functions f and g such that \(f(x)+f(x−1)−1=0\) and \(g(x)=x^2\). If \(f(x^2−x)=5\),then the value of the sum \(f(g(5))+g(f(5))\) is
- The functions f(x) and g(x) are related as f(g(x)) = xg(f(f(x))), where f (x) = \(\frac{x}{x-1}\). What could be the functional form g(x)?
Questions Asked in CAT exam
- For any natural Number 'n', let an be the largest number not exceeding \(\sqrt{n}\) , then a1 + a2 + a3... +a50 =
- CAT - 2024
- Number Systems
- The number of all positive integers up to 500 with non-repeating digits is
- CAT - 2024
- Number Systems
- The sum of all real values of $k$ for which $(\frac{1}{8})^k \times (\frac{1}{32768})^{\frac{4}{3}} = \frac{1}{8} \times (\frac{1}{32768})^{\frac{k}{3}}$ is
- CAT - 2024
- Logarithms
- When $10^{100}$ is divided by 7, the remainder is ?
- CAT - 2024
- Basics of Numbers
- In September, the incomes of \(K\), \(A\), and \(V\) are in the ratio \(8:6:5\). They rent a house, contributing \(15\%\), \(12\%\), and \(18\%\) of their respective incomes. In October, the rent remains the same, but the salaries of \(K\), \(A\), and \(V\) increase by \(10\%\), \(12\%\), and \(15\%\), respectively. What percentage of their total income is paid as house rent in October? (Round to the nearest percentage.)
- CAT - 2024
- Percentages
View More Questions