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- Compute the second derivative of f(x)=ex+x at x = 1 using a step size h = 0.2Shows that if a function f(x) is continuous in interval [a,b] then most likely the derivative of f(x) is also continuous at interval [a,b]When the Second Derivative Test Fails Analyze the critical points off (x, y) = 3xy 2 - x 3 .
- Estimate approximate derivative of f(x) = x2 at x = 1, for h = 0.2, 0.2, 0.05, 0.01 using first order forward difference formulaFind a function that is differentiable at every point of an interval [a,b] but that the third derivative does not exist at any pointUse the Chain Rule to express the second derivative off o gin terms of the first and second derivatives of f and g.
- The function f(x) is differentiable on (0, 1) and satisfies f(0) = f(1). However, its derivative is never zero on (0, 1). Does this contradict Rolle’s Theorem? Explain.Give an example of a function that is continuous on −∞ < x < ∞, but has NO DERIVATIVE at x = 3.What is the numerical approximation of the derivative of the function f(x) = x^3 - 2x + 1 at x = 2 using the forward difference formula with a step size of h = 0.1?