Engineering Mat...

The value of ‘C’ of the Cauchy’s mean value theorem for f(x) = e^x and g(x) = e^-x in [2, 3] is _____.

Consider a function g(x) = x.cos x that satisfies the following equation g”(x) + g(x) + k sin x = 0. What is the value of k?

Given the following statements about a function \(f:R\rightarrow R\), select the right option:

P: If \(f(x)\) is continuous at \(x=x_0\), then it is also differentiable at \(x=x_0\)

Q: If \(f(x)\) is continuous at \(x=x_0\), then it may not be differentiable at \(x=x_0\)

R: If \(f(x)\) is differentiable at \(x=x_0\), then it is also continuous at \(x=x_0\)

Consider a function f(y) = y³ - 7y² + 5 given on interval [p, q]. If f(y) satisfies hypothesis of Rolle’s theorem and p = 0 then what is the value of q?

If \(f\left( x \right) = {x^3} - 3x - 1\) is continuous in the closed interval \(\left[ {\frac{{13}}{7}, - \frac{{11}}{7}} \right]\) and f’(x) exists in the open interval \(\left( {\frac{{13}}{7}, - \frac{{11}}{7}} \right)\) then find the value of c such that it lies in \(\left( {\frac{{13}}{7}, - \frac{{11}}{7}} \right)?\)

What is the minimum value on the interval [4, 5].of the below given function?

f(x) = 3x3 – 40.5x2 + 180x + 7

Let the function \(f\left( \theta \right) = \left| {\begin{array}{*{20}{c}}{\sin \theta }&{\cos \theta }&{\tan \theta }\\{\sin \left( {\frac{\pi }{6}} \right)}&{\cos \left( {\frac{\pi }{6}} \right)}&{\tan \left( {\frac{\pi }{6}} \right)}\\{\sin \left( {\frac{\pi }{3}} \right)}&{\cos \left( {\frac{\pi }{3}} \right)}&{\tan \left( {\frac{\pi }{3}} \right)}\end{array}} \right|\)

Where \(\theta \in \left[ {\frac{\pi }{6},\frac{\pi }{3}} \right]\) and f’(θ) denote the derivative of f with respect to θ. Which of the following statements is/are TRUE?

(I) There exists \(\theta \in \left( {\frac{\pi }{6},\frac{\pi }{3}} \right)\) such that f’(θ) = 0

(II) There exists \(\theta \in \left( {\frac{\pi }{6},\frac{\pi }{3}} \right)\) such that f’(θ) ≠ 0

Find the value of ‘c’ lying between a = 0 and b = ½ in the Mean Value Theorem for the function f(x) = x(x - 1)(x - 2)

If f(x) is differentiable and g’(x) ≠ 0 such that f(1) = 4, f(2) = 16, f’(x) = 8g’(x) and g(2) = 4 then what is the value of g(1)?