Problem 11 Let f(x) be a differentiable function...

Problem 11 Let $f(x)$ be a differentiable function satisfying $$\lim_{x \to 1} \frac{2f(x) - 1}{x - 1} = 4 \quad \text{and} \quad \lim_{x \to \frac{1}{2}} \frac{f(x)}{2x - 1} = 1.$$ Let $g(x) = \frac{f^{-1}(x)}{x}$, where $f^{-1}$ represents the inverse function of $f$. Compute the slope of the tangent line to $g(x)$ at $x = 1/2$. (a) -3 (b) -1 (c) -3/4 (d) -1/2 (e) Cannot be determined

Transcript

00:01 We have been given to functions fx and gx and their derivative at the point one to three four now we have to find h -dash one h x is given by x is given by xfx plus 4gx this is the first term so to obtain h -dash -x we will differentiate this function with respect to x so ddx of x fx thus 4g x this is given by we will use product rule of differentiation to differentiate this function x multiplied by fx we will consider x as the first function if it is the second function so derivative of x is 1 1 times fx plus x times f dash x and the derivative of g x will be g -dhax therefore h -1 will be equal to f1 plus 1 times f -1 plus 4g dash 1 now the value of f1 as given in the table is 20 and the value of f -dash -1 is given as minus 12 dfdx at 1 is minus 12 and the value of g -1 is given as minus 12 so this is equal to 20 minus 12 minus 48 that is 20 minus 60 or it is equal to minus 40 so the value of h -dash -1 is minus 40 now in the second part hx is given as fx by gx and we need to find h -dash 2 so we have to again differentiate hx with respect to x so h -dash x is equal to ddx of fx g x now we need to use the question formula of derivative to differentiate this one fx by gx so the derivative is given by f -d -x gx minus the first function multiplied by derivative of the second function by gx square so h -dash -2 will be given by we will substitute two for x so f -dash -2 g2 minus f -2 g -2 g -2 by g2 square this is minus minus minus 8 plus 30 by 1 that is equal to 22 in the third part of the sum h x is given as twice x plus fx times g x now we'll again differentiate this function so h -taz x is equal to ddx of twice x plus f x times of g x and the derivative is equal to 2 plus f -dash -x g x plus fx g -x g -x so we need to find h -dash -3 and h -dast 3 is given by 2 plus f -dash 3 g 3 plus f3 g -3 we will use the table to substitute these values the value of f -3 is minus 4 g3 is 0 f3 is 4 and g -dess 3 is also 0 so the value is given by 2 and for the fourth part hx is given by 1x plus g x by fx now we will differentiate hx so ddx of 1 by x plus g x by f x and the differentiation is given by minus 1 by x squared plus g dash x fx x x minus g x f dash x by fx square...

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