existence condition

Practise existence condition with an original VectorPaths question, worked explanation and related VCE Methods practice.

2 practice marks

Original practice question

Let $h(x) = e^{2x} - 6e^x + 5$, where $x \in R$. The function $h(x)$ has exactly one stationary point, a local minimum. Find the largest value of $b$ such that when $h$ is restricted to the interval $(-\infty, b]$, the restricted function has an inverse.

Worked explanation

• Find stationary point

  \[h'(x) = 2e^{2x} - 6e^x = 0 \implies e^x = 3 \implies x = \ln 3\]

• Determine largest b

  \[b = \ln 3\]

$b = \ln 3$

Original VectorPaths practice; not an official VCAA question or marking rubric.