Let be a twice–differentiable function whose first derivative is given by h'(x)=3 x -…

MATHEMATICS · Multi-part · A Level · Mathematics · Integration · Integration

Problem

Let

h

be a twice–differentiable function whose first derivative is given by

h'(x)=3\sqrt{x}-\frac{2}{x^{2}}+7\sin x+e^{5x}, \qquad x>0.

It is known that

h(1)=4

. (a) Determine the general expression for

h(x)

. (b) Find the particular form of

h(x)

that satisfies the given initial condition. (c) Using your result in part (b), write the equation of the tangent line to the graph of

h

x=1

. (d) Show that

h''(x)=\dfrac{3}{2x^{1/2}}+\dfrac{4}{x^{3}}+7\cos x+5e^{5x}

and briefly interpret the sign of

h''(1)

in terms of the concavity of the graph at

x=1