class 12

Math

Calculus

Application of Derivatives

Let $f[0,1]→R$ (the set of all real numbers be a function.Suppose the function f is twice differentiable, $f(0)=f(1)=0$,and satisfies $f_{′}(x)–2f_{′}(x)+f(x)≤e_{x},x∈[0,1]$.Which of the following is true for $0<x<1?$

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Find the slope of the tangent to the curve $y=3x_{4}−4x$at $x=4$.

Find the least value of a such that the function f given by $f(x)=x_{2}+ax+1$is strictly increasing on $(1,2)˙$

Find the maximum and minimum values of f , if any, of the function given by $f(x)=∣x∣,x∈R$.

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Find local maximum and local minimum values of the function f given by$f(x)=3x_{4}+4x_{3}−12x_{2}+12$.